See NFHS 5.8.3D.

And?

Let's Go To The Videotape ...
 

5.8.3 SITUATION D: A1 or A2 requests a time-out: (a) while airborne A1 is
holding the ball; (b) while A1’s throw-in is in flight toward A2; or (c) when the ball
is on the floor at A1’s disposal for a throw-in. RULING: The request is granted in
(a) and (c), but denied in (b), as there is no player control while the ball is loose
between players.

Timeout ...
 

Because the request wasn't made when A2 caught the ball, it was made while the ball was in flight, which was denied (not granted) by the official. The request of timeout that's denied doesn't carryover over until the next opportunity to grant such a request. It takes a new request, when A2 catches the ball, which can then be granted by the official.

"If he makes this shot I'd like a TO ref." I'm not telling him to ask me again after a made shot.

I do like your explanation, kind of... my issue is the coach was asking multiple times. My question was to Bainsey's post and not the OP. I wanted a clear answer why he didn't give a TO when A2 had the ball and all I got was a rule that didn't tell me anything about what happened. I can see a coach asking for a TO on an interrupted dribble situation where he's praying an official kicks the rule to stop a potential steal. Yeah, of course I'm not even paying attention to the request. On the other hand, if A1 and A2 are by themselves in the BC and the coach says he'd like a TO but it's during a pass then I'm waiting for A2 to catch the ball and giving it to him.

If the coach keeps asking for a TO I'm not going to ignore him because it was a split second before a player caught the ball.

Can't Request A Time Out During A Live Ball Unless ...
 

I am. "Coach, I'll be sure to listen for you to yell time out when you want it."

I'm sure that many will disagree, but that's the way I roll, and I haven't had a coach complain about it in thirty-five years.

5-8-3: Time-out occurs and the clock, if running, shall be stopped when an official: Grants a player’s/head coach’s oral or visual request for a time-out,
such request being granted only when:
a. The ball is at the disposal or in control of a player of his/her team.
b. The ball is dead, unless replacement of a disqualified, or injured player(s),
or a player directed to leave the game is pending, and a substitute(s) is
available and required.

What If ...
 

What if it's a half of a second? A full second? Two seconds? Five seconds? What if the pass is ever so slightly deflected by a defender, but is still caught by A2? What if A2 fumbles the catch and it rolls around on the floor for a "split second"?

5-8-3: Time-out occurs and the clock, if running, shall be stopped when an official: Grants a player’s/head coach’s oral or visual request for a time-out,
such request being granted only when:
a. The ball is at the disposal or in control of a player of his/her team.
b. The ball is dead, unless replacement of a disqualified, or injured player(s),
or a player directed to leave the game is pending, and a substitute(s) is
available and required.

I'm not disagreeing and maybe your method is better than mine, but I've never mucked up a TO at any level -- I'm sure I will now though. There are just some situations I think it's a judgement call.

B1 shoots and scores. All B players run back on defense. A1 passes the ball in to A2 and the closest B player is 50 feet away. If the coach says TO then turns around to pickup his clipboard or starting talking with his bench, I have no problem giving him the TO when A2 gets the ball.

Granted, not requested is how I understand it. Hence, personally, I'm okay with a coach saying, "If he makes this FT I want a TO."

Different ...
 

Not better. Not worse. Just different.

Who am I to judge? (Pope Francis)

I've done this. Now, however, when a coach tells me this, I try to always take a look at them and make sure they still want it.

Specific guidance in NCAAW to have the coach ask again after the FT. It is what I've used at all levels.

Due diligence. I agree.

Have you ever had problems doing it this way?