On the other hand, setting the human background allowed them to do some interesting things with the PC's family's involvement that they wouldn't have been able to otherwise. There's an argument for doing it again for a similarly personal plot, though I generally prefer to have race choice.
Yeah that's true and you raise a good point. I guess I just never really felt overly invested in the Hawke family, so the specialness of the specific tailoring that being a "human only" allowed was kinda lost on me. I felt more attached to the families we got to meet in Origins than I ever did to the Hawke family. Apart from the Champion, I only really liked/cared about Carver, but since I primarily play a rogue, he's not in most of my games sadly.
It's a personal thing I know and I also know a lot of people really loved Hawke and his/her family (I've seen plenty of people comment on Leandra's death being especially hard to bear while playing) but I think Bioware is capable of making the story feel involved and relevant with every race. Couslands had the whole Landsmeet + Arl Howe ordeal, the mages had the Circle of Magi, the city elf had the Alienage storyline, the dwarves had Orzammar (and this one was easily my favourite). The only origin that kinda got the shaft in that department was the Dalish elf but even then they had Tamlen's encounter later and some special interaction with Zathrian's clan.
By contrast how they dealt with the Inquisitor's background and making it relevant to the overall plot felt a bit shallow by Origins and even DA2's standards, though I loved the race reactivity. It felt like being a dwarf/elf/qunari Inquisitor actually mattered, at least in some circumstances (i.e. war table missions, dialogue, court approval, romance choice, etc). But that's a topic for another time, I'm reaching TL;DR status! 