I always got the impression that, despite their financial issues, the Weasleys were too proud for “handouts.”

Yeah, that’s pretty much stated outright in GoF 10:

Harry looked away. He would willingly have split all the money in his Gringotts vault with the Weasleys, but he knew they would never take it.

If Harry had started buying the Weasleys a bunch of stuff, it would have just made the entire situation more awkward.