If your SPA routing is not updating the URL and title of your page, your router is garbage and that's on you, not on SPAs. A SPA must and easily can update your URL, title and tab history with minimal effort nowadays. That's what History API was made for.
You can break accessibility in your site with only a line of CSS. It is not a matter of tools, it is a matter of knowledge and care.
A SPA can be accessible, fast and a joy to users and developers. But you have to make it right and use when it makes sense.