QSettings не сохраняет информацию после закрытия программы

Я немного изменил ваше приложение:

• перешел с виджета QTableView на виджет QTableWidget;
• реализовал свою сортировку по столбцу "Points" используя класс QStyledItemDelegate;
• добавил функциональность по нажатию кнопки "Sort";
• реализовал сохранение настроек приложения используя класс QSettings.

QSettings - это абстракция позволяющая вам сохранять и восстанавливать параметры приложения в портативном режиме. Он также поддерживает пользовательские форматы хранения. Я добавил некоторую раскраску ячеек таблицы, когда значение point больше 777, для того, чтобы продемонстрировать сохранение настроек, в том числе и цвета ячейки.

Объекты, задействованные в сохранение настроек, помечены комментарием # QSettings. Пробуйте, но помните, что порядок написанных строк - имеет значение:

import sys
import contextlib
from random import randint
from PyQt5 import QtWidgets, QtCore, QtGui
from PyQt5.Qt import *


class SettingsManager:                                                           # QSettings
    def __init__(self, filename):
        self.m_settings = QtCore.QSettings(filename, QtCore.QSettings.IniFormat)

    @property
    def settings(self):
        return self.m_settings

    def read(self, widget):
        self.settings.beginGroup(widget.objectName())
        if isinstance(widget, QtWidgets.QAbstractItemView):
            selectionMode = self.settings.value(
                "selectionMode", type=QtWidgets.QAbstractItemView.SelectionMode
            )
            widget.setSelectionMode(selectionMode)

        if isinstance(widget, QtWidgets.QTableWidget):
            rowCount = self.settings.value("rowCount", type=int)
            columnCount = self.settings.value("columnCount", type=int)
            widget.setRowCount(rowCount)
            widget.setColumnCount(columnCount)
            items = self.settings.value("items")
            if items is None:
                self.read_defaults(widget)
            else:
                stream = QtCore.QDataStream(items, QtCore.QIODevice.ReadOnly)
                while not stream.atEnd():
                    it = QtWidgets.QTableWidgetItem()
                    i = stream.readInt()
                    j = stream.readInt()
                    stream >> it
                    widget.setItem(i, j, it)
                selecteditems = self.settings.value("selecteditems")
                stream = QtCore.QDataStream(
                    selecteditems, QtCore.QIODevice.ReadOnly
                )
                while not stream.atEnd():
                    i = stream.readInt()
                    j = stream.readInt()
                    it = widget.item(i, j)
                    if it is not None:
                        it.setSelected(True)
        self.settings.endGroup()

    def write(self, widget):
        self.settings.beginGroup(widget.objectName())
        if isinstance(widget, QtWidgets.QAbstractItemView):
            self.settings.setValue("selectionMode", widget.selectionMode())

        if isinstance(widget, QtWidgets.QTableWidget):
            self.settings.setValue("rowCount", widget.rowCount())
            self.settings.setValue("columnCount", widget.columnCount())
            items = QtCore.QByteArray()
            stream = QtCore.QDataStream(items, QtCore.QIODevice.WriteOnly)
            for i in range(widget.rowCount()):
                for j in range(widget.columnCount()):
                    it = widget.item(i, j)
                    if it is not None:
                        stream.writeInt(i)
                        stream.writeInt(j)
                        stream << it
            self.settings.setValue("items", items)
            selecteditems = QtCore.QByteArray()
            stream = QtCore.QDataStream(
                selecteditems, QtCore.QIODevice.WriteOnly
            )
            for it in widget.selectedItems():
                # print(it.row(), it.column())
                stream.writeInt(it.row())
                stream.writeInt(it.column())
            self.settings.setValue("selecteditems", selecteditems)
        self.settings.endGroup()

    def release(self):
        self.m_settings.sync()

    def read_defaults(self, widget):
        if  widget.objectName() == "tableWidget":
            widget.setSelectionMode(QtWidgets.QAbstractItemView.MultiSelection)
            widget.setRowCount(0)     #(1)
            widget.setColumnCount(2)


@contextlib.contextmanager                                                       # QSettings
def settingsContext(filename):
    manager = SettingsManager(filename)
    try:
        yield manager
    finally:
        manager.release()


class Delegate(QStyledItemDelegate):
    colorChanged = pyqtSignal(int, int, int)
    
    def __init__(self, parent=None):
        QStyledItemDelegate.__init__(self, parent)
        self.parent = parent

    def createEditor(self, parent, options, index):
        if index.column() == 1:
            value = index.data()
            editor = QLineEdit(parent)
            return editor        

    def setEditorData(self, editor, index):
        if index.column() == 1:
            value = index.model().data(index, Qt.EditRole)
            try:    value = int(value)
            except: value = 0
            editor.setText(f'{value:>20}')   

    def updateEditorGeometry(self, editor, options, index):
        editor.setGeometry(options.rect)

    def setModelData(self, editor, model, index):
        if index.column() == 1:
            try:    value = int(editor.text())
            except: value = 0
            value = f'{value:>20}'  
            model.setData(index, value, Qt.EditRole)
            self.colorChanged.emit(index.row(), index.column(), int(value))  


class Counter(QtWidgets.QMainWindow):
    def __init__(self):
        super().__init__()
        central_widget = QtWidgets.QWidget()
        self.setCentralWidget(central_widget)

        self.tableWidget = QTableWidget(0, 2)
        self.tableWidget.setObjectName("tableWidget")
        self.tableWidget.setHorizontalHeaderLabels(["Name" , "Points"])
        self.tableWidget.horizontalHeader().setSectionResizeMode(QtWidgets.QHeaderView.Stretch)
        
        delegate = Delegate(self)
        delegate.colorChanged.connect(self.color_changed)
        self.tableWidget.setItemDelegate(delegate)     

        update_button = QtWidgets.QPushButton("Update")                           
        update_button.clicked.connect(self.on_update_button)                      
        self.sort_button = QtWidgets.QPushButton("Sort")
        self.sort_button.setCheckable(True)
        self.sort_button.toggled.connect(self.on_sort_button)  

        self.qlineedit_name = QtWidgets.QLineEdit()
        self.qlineedit_name.resize(24, 80)
        self.qlineedit_points = QtWidgets.QLineEdit()
        self.qlineedit_points.resize(24, 80)

        horisontal_layout = QtWidgets.QHBoxLayout()
        horisontal_layout.addWidget(self.qlineedit_name, stretch=1)
        horisontal_layout.addWidget(self.qlineedit_points, stretch=1)
        horisontal_layout.addStretch(1)
        horisontal_layout.addWidget(update_button)                                 
        horisontal_layout.addWidget(self.sort_button)
        horisontal_layout.setAlignment(QtCore.Qt.AlignRight)

        grid_layout = QtWidgets.QGridLayout(central_widget)
        grid_layout.addLayout(horisontal_layout, 0, 0)
        grid_layout.addWidget(self.tableWidget, 1, 0)  
       
        self.read_settings()                                                     # QSettings
        
        self.tableWidget.setSortingEnabled(True)                     
        self.tableWidget.sortItems(1, Qt.DescendingOrder)

    def color_changed(self, row, col, value):
        if value > 777:
            color = QColor(qrand() % 256, qrand() % 256, qrand() % 256)
            self.tableWidget.item(row, col).setBackground(color)
            self.tableWidget.item(row, col).setFont(QFont("Consolas", 10, QFont.Bold))
        else:
            self.tableWidget.item(row, col).setBackground(QColor("#fff"))
            self.tableWidget.item(row, col).setFont(QFont("Consolas", 10))         

    def on_update_button(self):
        name = self.qlineedit_name.text().strip()
        point = self.qlineedit_points.text().strip() if self.qlineedit_points.text().strip() else '0'
        if not point.isdigit():
            msg = QtWidgets.QMessageBox.information(
                    self, 'ВНИМАНИЕ', 'Заполните правильно поле ввода Points!')
            return msg
        if not name:
            msg = QtWidgets.QMessageBox.information(
                    self, 'ВНИМАНИЕ', 'Заполните поле ввода Name!')
            return msg
        self.tableWidget.setSortingEnabled(False)
        rows  = self.tableWidget.rowCount()
        add_record = True

        for row in range(rows):
            if name == self.tableWidget.item(row, 0).text():
                #print(name)
                add_record = False
                row_edit = row
                break

        if add_record:       
            if self.tableWidget.selectedIndexes():  
                row = self.tableWidget.selectedIndexes()[-1].row()
                self.tableWidget.insertRow(row+1)
                self.tableWidget.setItem(row+1, 0, QTableWidgetItem(name))      
                self.tableWidget.setItem(row+1, 1, QTableWidgetItem(f'{point:>20}'))
                self.color_changed(row+1, 1, int(point))
            else:
                self.tableWidget.insertRow(rows)
                self.tableWidget.setItem(rows, 0, QTableWidgetItem(name))       
                self.tableWidget.setItem(rows, 1, QTableWidgetItem(f'{point:>20}'))
                self.color_changed(rows, 1, int(point))
        else:                 
            self.tableWidget.setItem(row_edit, 1, QTableWidgetItem(f'{point:>20}'))
            self.color_changed(row_edit, 1, int(point))

        self.qlineedit_name.clear()
        self.qlineedit_points.clear()      
        self.tableWidget.setSortingEnabled(True)
        
    def on_sort_button(self, state):
        if state: self.tableWidget.sortItems(1, Qt.DescendingOrder)
        else:     self.tableWidget.sortItems(1, Qt.AscendingOrder)

    def closeEvent(self, event):                                                 # QSettings
        self.tableWidget.setSortingEnabled(False)    
        self.write_settings()
        super().closeEvent(event)

    def read_settings(self):                                                     # QSettings
        with settingsContext("data_table_widget.ini") as m:
            for children in self.findChildren(QtWidgets.QWidget):
                if children.objectName():
                    m.read(children)

    def write_settings(self):                                                    # QSettings
        with settingsContext("data_table_widget.ini") as m:
            for children in self.findChildren(QtWidgets.QWidget):
                if children.objectName():
                    m.write(children)


if __name__ == "__main__":
    app = QtWidgets.QApplication(sys.argv)
    app.setFont(QtGui.QFont("Consolas", 10)) 
    app.setStyle('Fusion')                     
    w = Counter()
    w.setWindowTitle("Counter")
    w.setMinimumSize(480, 380)
    w.show()
    sys.exit(app.exec_())