Ugh. I didn't find anything on this issue a few weeks ago, so I didn't think to search again before posting this today. I just realized that this same issue is being discussed in the thread MWG cannot make SSL-Handshake with www.iif.com. Unfortunately what appears to be a fix in 126.96.36.199 is a setting not yet available to me in 7.6.2, so I'll look through the release notes of 7.6.x updates to see if it's covered anywhere yet.
I guess the point is to fallback also to TLS1.0. This help us even on 188.8.131.52
So first, try to handshake with TLS1.2 and TLS1.1 and TLS1.0. When handshake fails, try to fallback again to TLS1.0 :-)