DAO was near total trash with races. It NEVER mattered save those 3 insults -- and I'm not kidding that is it. Elves for lack of a better comparison sit on the same social tier as black Americans in the Jim Crow era. There is no way you would imagine a story about a character in 1930's America being basically the same with a white character and a black character. If you don't want to have to deal with the implications of race then do the Baldur's Gate world where race means nothing and no one even seems to care, or TES. The problem for Bioware is that in both DA and ME they made those choices means something and so there are consequences and differences.
...and yes Shep being human mattered. In ME1 he was an outsider both in terms of the council but also the council races and their pasts and history. He could for example interact with Wrex in a way a Turian couldn't. In ME2 again the humans are marginalized by the council and the pro-human radicals needs, say, a human. This is likely, oddly for a game about saving humans, the most fungible game in terms of his race. By 3, it still matters. The whole genophage cure thing would have to have played out wildly differently if Shep was a Krogan or Salarian...unless again "different" just means someone say "I don't trust salarians" you saying "You have to" and the conversation going on same as any other race.
Your race did matter in DAO more than just a few insults. If you played as a dwarf, people treated you like scum when you went back to Orzammar (Either for being a "traitorous" outcast or a casteless), Zathrian and his clan were more inviting if you were Dalish, You got treated better if you were human, You could'nt become Queen-Consort if you were anything but human, You got a heroes welcome in the Alienage if you were a city-elf, and I'm sure I'm not even close to pointing it all out.
Also, the race you played as could make you interpret your playthrough differently. A human HoF might not feel too bad about taking out the Dalish clan and siding with the werewolves, but a Dalish HoF would probably be vehemently opposed to doing that (Depending upon how you were playing your character of course)
And two things that you said really made me disappointed in the lack of playable races in ME
"He could for example interact with Wrex in a way a Turian couldn't" How awesome it could have been to interact with squadmates as different races? Like interacting with Legion as a Quarian Shepard
"The whole genophage cure thing would have to have played out wildly differently if Shep was a Krogan or Salarian" A Krogan-Shepard would be VERY interested in finding a cure for the genophage and a Salarian-Shepard might feel guilt at being the race that inflicted such a disease upon the Krogans (Again, open to how you want to play your character)