It depends how websites choose to implement it, and how other browsers choose to implement it.

If Firefox et.al chooses not to implement browser environment integrity, then any website that chooses to require strict integrity would completely cease to work on Firefox as it would not be able to respond to a trust check. It is simply dead. However, if they do implement it, which I imagine they would if this API actually becomes widespread, they should continue to work fine even if they're stuck with the limitations on environment modification inherent to the DRM (aka rip adblockers)

Websites will vary though. Some may not implement it at all, others may implement a non-strict integrity check that may happily serve browsers that do not pass the check. Third parties can also run their own attestation servers that will report varying levels of environment data. Most likely you will see all Google sites and a majority of "big" websites that depend on ad revenue implement strict integrity through Google attestation servers so that their precious ads don't get blocked, and the internet will become an absolutely horrid place.

Frankly I'll just stop using anything and everything that chooses to implement this, since we all know Google is going to go full steam ahead with implementation regardless of how many users complain. Protecting their ad revenue is priority 1 through 12,000 and fuck everybody else.