Thanks to @musicamante, I realized I had to supply my own html template to export_html. That was not immediately clear to me as its named a bit weird. Not only was I injecting entire HTML pages into the TextEdit, but also the <pre> tags that are used by default cause the spacing issues, so i used <div> instead.

I also set the QTextEdit to use the monospace font that would normally be selected in the HTML stylesheet.

This is what ended up working:

class QTextEditLogger(QTextEdit, Handler):
    """A QTextEdit logger that uses RichHandler to format log messages."""
    def __init__(self, parent=None, level=NOTSET):
        QTextEdit.__init__(self, parent)
        Handler.__init__(self,level=level)
        self.console = Console(file=open(os.devnull, "wt"), record=True)
        self.rich_handler = RichHandler(show_time=False, show_path=False, show_level=True, markup=True, console=self.console, level=self.level)
        self.rich_handler.setLevel(self.level)
        self.setWordWrapMode(QTextOption.WrapMode.WordWrap)
        self.setAcceptRichText(True)
        self.setReadOnly(True)
        font = QFont(['Menlo','DejaVu Sans Mono','consolas','Courier New','monospace'], 10, self.font().weight())
        font.setStyleHint(QFont.StyleHint.TypeWriter)
        self.setFont(font)


    def emit(self, record) -> None:
        """Override the emit method to handle log records."""
        self.rich_handler.emit(record)
        html_template = '<div style="background-color: {background}; color: {foreground};><code style="font-family:inherit">{code}</code><br/></div>'
        html = self.console.export_html(clear=True, code_format=html_template, inline_styles=True)
        self.insertHtml(html)
        self.verticalScrollBar().setSliderPosition(self.verticalScrollBar().maximum())  
        c = self.textCursor()
        c.movePosition(QTextCursor.End)
        self.setTextCursor(c)