You're not even giving a reason why they shouldn't, though.
I gave the reason, you just disagreed with it.
Dragon Age is a perfect example, and I did not in fact limit my example to DAI. It is largely the same in Dragon Age Origins, except you get a unique prologue. This isn't so different from ME1 in which your backgrounds gave you different minor sidequests and some small changes to dialogue. DAO is basically the same game, same story, played as any race. This is the biggest example of racial and background differences in ME or DA and it is pretty minor.
"Not all of the game happens in the game" - actually yes, all of the game does in fact occur in the game.
I disagree that Bioware should waste effort in the development of allowing multiple playable races. It is unlikely that each race would actually play differently in any meaningful manner. It does not make sense to ask them to spend time developing race specific dialogue when they had a history of dumbing down the dialogue wheel over the course of the ME series.
I actually don't mind if they shoehorn in multiple races just so some random person can play Asari for the sake of playing an Asari. As long as it doesn't steal development time from balancing the core classes and making meaningful differences and tones available in conversations with a given character.