There are no good reasons to play 1. h3 (or 1. a3) other than psychology (confusing the opponent, avoiding theory/opponent's preparation/...).
Regarding the two "reasons" you mention:
Gives white a chance to play like black (respond to an opening rather than create one)
I don't see this as an argument for playing a waiting move like 1. h3/a3. Why would you want to play like black (which has statistically a lower winning percentage)? What is the advantage of "responding to an opening"? That would mean that you have to be prepared for all kind of moves, so if anything you would have more work, studying more lines, etc.
Also I disagree that it is always white who is "creating" the opening. In the end the opening is created by both players. Take for instance 1. d4. Black can respond with 1...d5 playing some kind of Slav or Queen's gambit which more often than not leads to symmetrical equalish positions. Or black can go for something like a King's Indian, Benoni, Grunfeld... which leads to asymmetric wilder play. So here it is black who decides the character of the opening.
Defends against bb4 [Bg4!?]
It is too early to know whether you want or need to defend against this. It is not like Bb4/Bg4 is a major threat (in most openings) that must be prevented. Also in some openings the bishop might not even want to (be able to) go to those squares. Lastly, pushing the pawn to h3/a3 can create weaknesses or targets for pawn storms in some lines.