Works on my side. Instead of a Text just use a SelectableText and it will work for you too. It is still a widget and doesn't behave like text would in an HTML page 100% but there's no reason why parity can't be achieved.
Also, there are constant improvements in the A11Y and SEO departments. You have quite a lot of control of how a widget is seen by the browser and what data to expose to the screen reader. When you build for the web depending on what you try to get out of the dart2js compiler you may use the --web-renderer CLI option with the html or canvaskit values.
Anyway, Flutter may live alongside the holly trinity (HTML/CSS/JS) and won't replace it. From my point of view, it is an option, like many others, and may work great in some scenarios to create a good UX and happy users. Used badly, it might create the exact opposite.