I love all these characters and I want to play as them!
(I can't imagine barbarian Gale for example, his story is strictly connected to his class). And even this would create problems - what if we can meet, let's say, Shadowheart's parents in the next acts? If you change her race, devs would have to think about it too - they would have to create EVERY possible combination of her parents - like drow + hight elf, dwarf + dwarf, human + tiefling etc..
So as to the first, noone is saying there cannot be basic pre requisites. To access an origin dealing with magic, one would obviously need to play certain classes. This is reasonable and presents no undue burden. As to the second. . .That is the sort of thing programming was made to do, execute basic if - then commands. If player is an Elf then parents equal Elves. Again, easy to implement and fail proof. This suggestion is incredibly easy to facilitate and there is really no reason not to use it. The only thing I see being an issue is voice work because then all male and female dialogues would have to be recorded by all voice actors. That would be a huge and discouraging cost. . .But ultimately I think it would be worth it to do something this liberating and unique. It really would be moving the state of the game forward.