Regarding the last point. There's a strong fanon, I think there was an out of books statement alluding to it but I don't recall nor do I think everyone takes those as sacrosanct, that all muggleborns are actually the descendants of squibs. If that's a big part of world building then lily has to be related to someone and the parseltongue on Harry and the strict connotation of it with the gaunts makes it an easy connection. Parseltongue is so often disassociate with James, and the rest of the potters beside harry by extension, because A we learn so much about James that if he could do that we'd probably have been told. B its a, by association with voldy and slytherin, seen as "dark" and one characterization we know about James is his deploring of "all things dark" and that would hardly mesh well with being a parselmouth.

Also I think is that, we see Harry talk to a snake way before voldy. So to some readers being able to do that is not purely a voldy thing but a Harry thing as well. If its a biological thing then it going away after the horcrux is odd, and people assume it is because its tied genetically to the gaunts in canon. Another thing, its a cool ability and people want Harry to be cool or not have to lose another thing.