I thought the issue was the OP's distaste for the word "race". Is people just as distasteful?
Every one of the OP's issues with the word race would apply to the word people as well. *shrugs* Its all basically a question of Political Correctness. Which I find to be odd, considering that PC terms are half the time more offensive than the term they replace.
The original point of Politically Correct terms was to avoid giving offense. However, more than half the time, the PC term was jsut as bad, if not worse, than the original. No one I know of is really offended by the term used in terms of fantasy racism, and its not attempting to insult anyone. Its more that some people are apparently unhappy with the connotations of race in D&D terms and in real life.* That said, however, "people" is often already charged with the same meaning and connotations as racism in reality. "We don't want your kind of people here!" "Ever since the bus route changed, there's more black people down at the store, and I'm scared to go there now!" Race and people are too intertwined to work.
* I think that, since different fantasy races are used as a stand in for different racial groups in real life / real world history, its actually more offensive to remove the term, as if its okay to think of the other races as less-than-human, and thus okay to slaughter. Part of the whole thing with fantasy is that most everything is a metaphor for something in reality, and we're talking about the human condition in some way. So, trying to distance yourself from that metaphor is a way to distance yourself from the actual bad things you do as a character in the game.
On the one hand, that's fine for those who are actually just there to play a mindless game to relieve stress as long as you don't associate. On the other, for something that's modeling the real world, it can have negative messages and unfortunate implications associated with it, as well as push away potential new gamers of various backgrounds. Toxic white-masculinity is harmful to the hobby in this age, especially for the younger kids who seem to be showing interest in the games.