Both are already happening.

Google markets Chrome relentlessly, with popups in search and YouTube if you're using other browsers, browser choice dialogs in Google iOS apps (despite iOS having a default browser setting for 5 years now), Chrome getting bundled into random Windows software installers, etc.

Many devs actively desire single-engine development and testing and many aren't shy about using Chrome only features already. If they had the capability to tell users to go install Chrome instead of targeting broadly supported features, they would do so in a heartbeat.

I have hit a few sites over the last year that threw up full page "This site only works in chrome" blocks, even though they usually work perfectly fine in Firefox if you set the user agent to chrome.