That list would be much larger than the list of sites that do require SNI.
Generally, I can determine whether SNI is required by IP address, i.e., whether it belongs to a CDN that requires SNI. Popular CDNs like AWS publish lists of their public IPs. I use TLSv1.3 plus ESNI with Cloudflare but they are currently the only CDN that supports it. Experimental but works great, IME.
The proxy maintains the list not the browser. The proxy is designed for this and can easily hold lists of 10s of 1000s of domains in memory. That's more domains than I visit in one day, week, month or year.
Is it not a question of whether this is possible. "How would this work". I have already implemented it. It works. It is not difficult to set up.
Why this works for me and would unlikely work for others.
I am not a heavy user of popular browsers, I "live on the command line". Installing a custom root certificate with appropriate SANs to suppress browser warnings is a nusiance that would likely dissuade others since they are heavy users of those programs. However I generally do not use those browsers to retrieve content from the web.