I recently had an alignment done, the car was pulling to the side (drifting to the side, not really a pull), had to take it back, have it aligned again, and it was fine with the second alignment. I was told the alignment was off by a tiny amount.
The drift is also more pronounced the faster you go.
I also replaced the tires before having the second alignment done. My old front tires were bald on the inside but looked almost brand new on the outside (never rotated), that's how badly screwed up they were, but replacing them didn't fix my drift. With old tires and new tires it drifted to the right.
So based on my experience I think it's unlikely that tires are the culprit. I agree with one of the suggestions, if you rotate the tires and it goes away it's the tires, if it doesn't it's something else.