One of the things that drives me nuts about the summer AAU basketball season is that, in order to keep tournaments on time, pre-game warm-ups are usually 3 minutes or less (even when running ahead of schedule). We know this going into every game, so we have the players do dynamic warm-ups and ball handling off-court.

The time is too short and valuable to have guys standing in line. We'd like to come up with a more efficient and structured way for our players to get up as many quality shots as possible during that time. What are other coaches doing besides just letting them shoot around?

Same here on all accounts!!

We tend to mix up our shooting.... but here are a few things we do:

Line at top of key, everyone has a ball except first player in line. First player cuts to right wing, receives pass, rips ball aggressively and attacks the basket for lay ups. Next player cuts to right. Working on rips and exploding to basket. It's a pretty fast paced because players alternate directions and then the coach gets them in a good rhythm to maximize reps.

For the same drill you can fake rip (jab step) and attack. You can also do pull up jump shots.

Sometimes we partner up and take 5-10 form shots. Approx 5 players around the basket taking shots without waiting.

Drive and kick shooting. 4 lines... guards lanes and pocket/wing area. Players in guard lane have a ball. The attack the middle and kick to shooter in opposite corner/pocket area. As soon as first player drives into lane, the next player drives into lane. Passer follows into same line. Shooter goes to passing line closest to them.

This is chaotic but gets maximum reps. You form two groups. Group one has 3 basketballs and 4 shooters, they take all shots on right half of court. Group 2 does same thing on left half of court.
https://www.breakthroughbasketball.com/play.asp?id=8032

Then there's the traditional line under the basket drill and the two lines on the wing drill. Those aren't bad IF you get players maximizing reps by moving right away and getting a really good rhythm. If you train players, it's not bad. If they do what they want, it is super slow... you can double the reps just by teaching them how to do it right.

I have yet to find the perfect drill... but those are all decent options. It's just not possible to get many shots in 3 minutes.