I think the issue is that you have a guy who min-maxed, if I understand correctly that he put all of his stat bonus points into DEX. If that is the case, then I get that he is rolling at a +6, which means he only needs a 9 on the die to trigger this ability, so no wonder the effect keeps triggering so frequently.
For the record, most people I play with do not assign stat points that way, as they are more interested in being well rounded team players than outshining their teammates.
Nonetheless, here are my thoughts on this issue. The challenge for you as a DM, then, is to find ways to challenge this player in a way that will allow your other players to also shine, without necessarily nerfing this player. In order to do so, I would do five key things.
One, saving throws are your friend, whether from enemies or the environment. If this guy has no points in str, con, int, wis, or cha, it’s time for him to have to feel that as a pain point. Having a high DEX won’t help him when a poison cloud rolls in, or if he is drowning/suffocating, or trying to resist extreme cold. Enemies and environments with these effects are huge. Next, when mind controlling illithids show up, this guy is a prime target. Make him make an int save to avoid turning on his friends. Suddenly, +6 DEX shots at your allies with quick draw isn’t such a good combo if he fails that save.
Two, varied enemy types are also your friend. Try throwing a rampaging Minotaur who is immune to all damage at the group, and the only way to damage it is to first climb on its back and remove an amulet. Or have them face a foe that absorbs all ranged attacks, like a corpulent zombie that explodes poisonous goo when it gets near. Go wild. Vary it up. Challenge your players in unique ways rather than just damage.
Three, create a nemesis for this player. A rival “sniper” that pings away at him. Oh, and that enemy also has a speed quiver or the quick draw ability. Or maybe that enemy has cloaking. He takes a shot, but then blinks or disappears. Now, your player has to make Wis rolls just to take a shot because he can’t find him. Super nuisance! Plus, it’s usually the case that opposing forces will target the PC posing the greatest threat. So, your guy might take down two bad guys, but three of his friends will target him. Or maybe he fights a mage with a force shield and the seekers spells. Nothing like making your player hide from auto-matic hitting seeking spell missiles!
Four, if the way this player shines is thinning the herd, time to lean into it. Let him shine and save the group. But let’s see how they do against 20 insectoid chub-chubs rushing them. Or 15 zombies pressing them in a tight hallway. With more than enough enemies to go around, everyone gets a chance to have a bite at the spotlight.
Five, don’t forget that hard rolls are your friend. Depending on the moment of the fiction, this player’s rolls might be hard. Unless he has a milestone or some other special ability, having a giant orc smashing against him in melee isn’t going to let him take normal shots. Or maybe the player is confined. Or maybe it’s all a long game against entrenched enemies who are using cover. Against such a skilled archer, smart enemies won’t throw themselves to their deaths out in the open, unless they are confident they can overwhelm him with superior numbers. They will use cover, retreat, or otherwise force that player to step up and make a more epic (hard) shot.
If those five pieces of challenge tuning don’t help you, then it’s probably just time for a conversation with him one on one. “Hey man, these other folks need a chance to shine. What suggestions do you have about quick draw being a little spotlight hogging?” Then get his ideas in terms of helping you. They’ll go down way better if he has a hand in the solution, plus the honest feedback.
Finally, as others have suggested, you can nerf the ability or put limitations on it. Among the options suggested, I would probably lean toward a limit on arrows or something bad happening on a Nat 1. Either the arrows are used up, or the bow string breaks, or the bow takes some effort damage. Two more Nat ones, and the wood will snap. But those would be my last resort, as the goal is for everyone to shine. Fooey on him for not building that way.
Also, as a side note. Seven players? Dang man. I’d split that group in two. You’re creating a ton of extra work for yourself trying to challenge a group that large and keep everyone engaged.