Can Search API filter "Index full entity" be used in Search API views?
This should be possible. I did the same thing with Search API + Elastic search (No Views module though). And yes it did speed up the results. However, in retrospect I may have been better of using Render Cache module in conjunction with an object cache like Redis and or Memcache.

The main pain point that I ran into doing full storage of HTML in the search server was that nearly every theming/template change required rebuilding the search index.

All that being said, if you want to keep going down this route I suggest using Solr's built in inspection tools to first validate that the user profile HTML has made it into Solr at all.