If you have a FIDO device on your (physical) keyring or a keyboard with a smart card reader or some kind of NFC transceiver connected to your PC, the problem is technically solved - just not practically.
Adding to this: anyone older than 12 years old is required by law to have their government issued ID on them at all times when in public. If your ID is suddenly your smartphone, you're essentially required to have that on you 24/7. Dystopian spyware.
Tragedy of the commons, nobody seems to have bothered to work on it. It's not like Chromium or Firefox wouldn't accept contributions.
Around a decade ago I was working at a company that used smartcard login for authenticating to internal sites. I've heard of many others doing the same. USB card reader worked fine in both IE and Firefox at the time, so I take your statement to mean that we've somehow regressed since then (not surprising) or this was an isolated instance of success (less likely, considering the US government also uses this: https://en.wikipedia.org/wiki/Common_Access_Card).
As an aside: signing things has a particularly awful UX. I never know what I'm signing, I have no way to verify that what's on the screen is what's being signed. And then there are orgs that use eID based PDF signing, which again requires different plugins. In short: a shitshow.