I'm less sure browsers wouldn't adopt, but if content providers are slow to adopt due to massive changes in how they handle sessions / tracking / everything else they do with cookies, browsers are not incentivized to adopt.

I suspect it would end up looking similar to IPv6: Unless ISPs are providing the majority of users with IPv6, software developers aren't well incentivized to support IPv6 in their software. Similarly to browsers, they'd lose very little for supporting the new standard now, but the gains are low while adoption by the other group is low.