Yeah, it is very close. But I feel simp has a bit of a sexual feel to it. Like a guy who does favors for a girl expecting affection in return, or donates a lot of money to an OnlyFans or Twitch streamer. I also see simp used where we used to call it white-knighting (e.g. "to simp for").

Obsequious is a bit more general. You could imagine applying it to a waiter or valet who is annoyingly helpful. I don't think it would feel right to use the word simp in that case.

In my day we would call it sucking up. A bit before my time (would sound old timey to me) people called it boot licking. In the novel "Catcher in the Rye", the protagonist uses the word "phony" in a similar way. This kind of behavior is universally disliked so there is a lot slang for it.

Thanks, as an old timer TIL about simp.