I don’t think that there is an obligation with that kind of standard, no.

Banking and security, accessibility yes.

Specific choice of “user side software”, probably not. And it’s somewhat unlikely to happen too, because if you think about apps on phones, if suddenly a completely new phone OS were to show up and had 30% market share, it wouldn’t make sense to have a law that would legally require them to offer an app on that platform

And Chrome isn’t “officially bad” in a legal sense.

The internet standards themselves are a bit… imprecise too. Implementing them in browser is ultimately up to the companies, there is no legal body requiring a browser to have or not have features. They just usually sort of do the same things because going different paths would be stupid. Mostly. Sometimes they totally do that, though, e.g. calendars and contact info have a standard, but all implementations are a mess and transfer is a pain.