Yes. You can use replace() instead of push().

But there are other issues like bookmarking or sharing the page with the menu open. People won't expect the result.

It's really not difficult to create an on-page menu so this seems more trouble than it's worth.

But would people really try to share / bookmark a page that at that very moment doesn't show the content they want to share?

I think it is a beautiful, simple solution that just works (and also did work perfectly in the past when we didn't have JS)

Yes. If you have ever worked on a site with lots of users, you will find that if there is a way to break something, someone will absolutely do that thing. It might not be super often, but it will definitely happen.