Hi,
I am afraid, Marshall, that you are mistaken about that; in fact, at Vivaldi (which is using Extended Stable) we generally start receiving reports issues about sites saying "You are using an obsolete browser, you must update to use this site" at least two weeks before the next Extended Stable is even available. Among sites where we think we have received such reports about, is Twitch, and I have myself seen lots of Cloudflare (and co) "Checking your browser" roadblocks once the Stable is two or three weeks old.
These sites are not using anything new. They are just blocking browsers based on their version numbers, calling them "outdated", "unsupported", "must update", etc..
Then there are sites like
https://www.whatismybrowser.com/ that seems unaware of the existence of Extended Stable.
This problem is so severe for our users that we are now starting to seriously consider the possibility of spoofing the current Stable Chromium version, that is, UA and Client Hints would e.g. for a v144 show v145 instead to web sites.
IMO, an (extreme, hacky) method that Chrome could use to help mitigate this situation would be to send (in UA and Client Hints) the Current Extended Stable Chromium version number in a significant (>10%) of installed Stable installations to make sure that the sites don't dare to block Extended Stable browsers.
Another way the Chrome/Chromium team can help is by overlapping patch support for the old Extended Stable for 4 weeks after a new Extended Stable is release; that will help downstream embedders that are releasing a week or two after the Extended Stable release