Using the code from Aaron as a base to start on, I managed to implement all the functionality required to work with my existing script:
from PySide2 import QtCore, QtWidgets
class TabBar(QtWidgets.QTabBar):
    def minimumSizeHint(self):
        """Allow the tab bar to shrink as much as needed."""
        minimumSizeHint = super(TabBar, self).minimumSizeHint()
        return QtCore.QSize(0, minimumSizeHint.height())
class TabWidgetPlus(QtWidgets.QWidget):
    tabOpenRequested = QtCore.Signal()
    tabCountChanged = QtCore.Signal(int)
    def __init__(self, parent=None):
        self._addingTab = False
        super(TabWidgetPlus, self).__init__(parent=parent)
        # Main layout
        layout = QtWidgets.QVBoxLayout(self)
        layout.setContentsMargins(0, 0, 0, 0)
        # Bar layout
        self._tabBarLayout = QtWidgets.QHBoxLayout()
        self._tabBarLayout.setContentsMargins(0, 0, 0, 0)
        self._tabBarLayout.setSpacing(0)
        layout.addLayout(self._tabBarLayout)
        self._tabBar = TabBar()
        self._tabBarLayout.addWidget(self._tabBar)
        for method in (
                'isMovable', 'setMovable',
                'tabsClosable', 'setTabsClosable',
                'tabIcon', 'setTabIcon',
                'tabText', 'setTabText',
                'currentIndex', 'setCurrentIndex',
                'currentChanged', 'tabCloseRequested',
            ):
            setattr(self, method, getattr(self._tabBar, method))
        self._plusButton = QtWidgets.QPushButton('+')
        self._tabBarLayout.addWidget(self._plusButton)  # TODO: Find location to insert
        self._plusButton.setFixedWidth(20)
        self._tabBarLayout.addStretch()
        # Content area
        self._contentArea = QtWidgets.QStackedLayout()
        layout.addLayout(self._contentArea)
        # Signals
        self.currentChanged.connect(self._currentChanged)
        self._plusButton.clicked.connect(self.tabOpenRequested.emit)
        # Final setup
        self.installEventFilter(self)
    @QtCore.Slot(int)
    def _currentChanged(self, i):
        """Update the widget."""
        if i >= 0 and not self._addingTab:
            self._contentArea.setCurrentWidget(self.tabBar().tabData(i))
    def eventFilter(self, obj, event):
        """Intercept events until the correct height is set."""
        if event.type() == QtCore.QEvent.Show:
            self.plusButton().setFixedHeight(self._tabBar.geometry().height())
            self.removeEventFilter(self)
        return False
    def tabBarLayout(self):
        return self._tabBarLayout
    def tabBar(self):
        return self._tabBar
    def plusButton(self):
        return self._plusButton
    def tabAt(self, point):
        """Get the tab at a given point.
        This takes any layout margins into account.
        """
        offset = self.layout().contentsMargins().top() + self.tabBarLayout().contentsMargins().top()
        return self.tabBar().tabAt(point - QtCore.QPoint(0, offset))
    def addTab(self, widget, name=''):
        """Add a new tab.
        Returns:
            Tab index as an int.
        """
        self._addingTab = True
        tabBar = self.tabBar()
        try:
            index = tabBar.addTab(name)
            tabBar.setTabData(index, widget)
            self._contentArea.addWidget(widget)
        finally:
            self._addingTab = False
        return index
    def insertTab(self, index, widget, name=''):
        """Inserts a new tab.
        If index is out of range, a new tab is appended.
        Returns:
            Tab index as an int.
        """
        self._addingTab = True
        tabBar = self.tabBar()
        try:
            index = tabBar.insertTab(index, name)
            tabBar.setTabData(index, widget)
            self._contentArea.insertWidget(index, widget)
        finally:
            self._addingTab = False
        return index
    def removeTab(self, index):
        """Remove a tab."""
        tabBar = self.tabBar()
        self._contentArea.removeWidget(tabBar.tabData(index))
        tabBar.removeTab(index)
if __name__ == '__main__':
    import sys
    import random
    app = QtWidgets.QApplication(sys.argv)
    test = TabWidgetPlus()
    test.addTab(QtWidgets.QPushButton(), 'yeah')
    test.insertTab(0, QtWidgets.QCheckBox(), 'what')
    test.insertTab(1, QtWidgets.QRadioButton(), 'no')
    test.removeTab(1)
    test.setMovable(True)
    test.setTabsClosable(True)
    def tabTest():
        name = 'Tab ' + str(random.randint(0, 100))
        index = test.addTab(QtWidgets.QLabel(name), name)
        test.setCurrentIndex(index)
    test.tabOpenRequested.connect(tabTest)
    test.tabCloseRequested.connect(lambda index: test.removeTab(index))
    test.show()
    sys.exit(app.exec_())
The one difference is if you're using tabWidget.tabBar().tabAt(point), this is no longer guaranteed to be correct as it doesn't take any margins into account. I set the margins to 0 so this shouldn't be an issue, but I also included those corrections in TabWidgetPlus.tabAt.
I only copied a few methods from QTabBar to QTabWidget as some may need extra testing.