They've already started. Google Meet (their version of Hangouts for enterprise) didn't work on non-Chrome browsers for a very long time. If you tried to do a video call with a non-Chrome browser, you were told to install Chrome and load the page in Chrome. The only reason they were able to get away with this is because Chrome has the high market share that it does. If they'd done this when Chrome had 10% market-share, the response from users would have been to stop using Google Meet. However, because they did this when Chrome has 70% market share (and climbing) there was hardly any outcry; those who didn't already have Chrome installed gave a resigned sigh and installed Chrome in order to participate in their work meetings.
Currently, all of Google's services work on non-Chrome browsers. But will that always be the case? I can very well imagine a world where Google starts making its services Chrome-only, citing, for example, that only V8 has the necessary Javascript performance to run Google's increasingly bloated webapps with adequate performance.
Like you said, it'll start as a slow drip, with the least popular and most obscure applications (for example: Google Play Music) being moved over first. Then, as users fail to object, they'll move over larger and larger applications. Even if they never make the "big-two" of YouTube and Google Search Chrome-only, they'll still be exerting a fair amount of pressure for users to switch to Chrome.