I wouldn't count on that. Web devs aren't going to push for this, it'll be the suits that have some dumb automated "security" tool tell them they need to enable it or they'll get hacked.
There will always be a cat and mouse game where some people figure out clever ways around this, but I wouldn't count on it being as easy as installing an addon. Sites could start requiring a specific attester that requires that you run their rootkit malware to spy on your entire OS and only supports a few popular OSes. Thanks to projects like TPM, your own hardware could be working against you.
As usual, Stallman predicted the world that large companies would like to drag us into: https://www.gnu.org/philosophy/right-to-read.en.html