I don't know that I've seen these errors with Web Client before, nor could I find any mention of them when I looked.
As Web Client is working fine on IE 11, I wouldn't think it a need for Web Client that TLS 1.0 be enabled or disabled, otherwise I wouldn't think it would work with any browser if that was the case.
If you installed Web Client from the MDGPFALL2020_DVD_ENUS media, it should function fine with TLS 1.0 and TLS 1.1 disabled, which sounds like what you're seeing, just with Firefox and Chrome, it doesn't work.
I'm not sure if there's something with those two browsers or not, I'd be curious if MS Edge browser works with Web Client for you or not?
I will say we haven't seen these errors yet with GP 18.3 and Web Client.
Thanks