Roles Can Be A Bummer
Making actions/options not available to people can force people out of the magic circle pretty quickly. Sure, roles can help people feel special, but it can go overboard. This does not mean that the rogue shouldn't be stealthy, or that the cleric can't be tank-y, but not being able to use or do something because of an arbitrary class restriction breaks the circle, which means less fun for everyone.
Sometimes, multiple people actually can do the same thing, and that should happen sometimes. If every task you throw at the player characters cannot be accomplished except for 1 person's specialization, it may be a harder adventure than these players (as a group) are fit for. Not that there can't be a situation where you need a particular player to do a something, but that it shouldn't happen all the time.
Perhaps It is a Matter of Perception
So maybe this player is simply just does not imagine the same thing you do. You can try to reinforce what you envision by describing what is going on. Is she/he making informed choices? Does she/he realize what the gravity of the situation, or how her/his character has certain strengths?
Does this player realize the cleric is running around in thick armor, while that player's character is running around in normal clothes? For example, if I think the cleric is running around in normal clothes, and goes into combat and comes out unscathed, I could run in my normal clothes, and also come out unscathed.
If this is the problem, you simply need to be more descriptive! Describe how the cleric is wearing armor, or how the rogue is especially cat-like, and how this player's character is not.
For Games Which Depend On Roles
It may be time to have your other players step up. Have the cleric yell at the other character to stay back, or that the cleric "has this." Alternatively, the cleric may ask "PEASE SUMMON THIS TO HELP ME!" as he charges in.
If the rogue attempts to sneak around, the rogue can give specific instructions; "Can you make a distraction over there?" "Wait here, and if I'm not back in 15 minutes, go get the others to rescue me." Or even: "this looks super dangerous. I don't think you'll make it. We don't want to get ourselves killed, yeah?"
In short, give him tasks, or have the other players give him tasks to help with things. Talking is a free action in combat; use it. This is especially good for players who may be experiencing some mental handicap; it gives them something concrete to act on. It allows them to contribute (which feels great and is fun) without going through negative experiences.
Finally, a player or the DM can specifically highlight when certain actions will require a specific specialization, and who has that training. If it becomes obvious that a task is dangerous and requires training, then most people leave it to the person with the training. The DM can further forbid people without specialization from trying, stating that it's obviously too hard. This is more "hand-holdy" than some people like it, but sometimes people just need their hands held.
Talk About It
Talk with the player about the roles. What is her/his character good at? What should she/he focus on? These other characters have something special about them, what is she/he special at?
Talk about how their character is so good at magic/whatever else, and how they should try to make their magic/whatever else be the solution for the current situation. (After all, we're solving these situations using our strengths, use your strengths to help solve it!)
Tough Love
Finally, you may have to resort to letting the consequences of this player's action happen. Let the events, despite other character's best efforts, happen. Talk about how that character was not focusing on what they were good at, or how their play-style does not match up well with the class they chose. Make a new character that does okay at everything, such as most games' version of a "bard." Note: this does not mean make a character who is rule-breaking and superior to all the others, but one that stands a decent chance at performing many things, and isn't a large risk to the success of most tasks.
Best Answer
I don't know your player, but I know something about playing D&D in AL with ADHD. That's my hobby. I also know something about working with young adults with ADHD in structured settings. That's my profession.
We shouldn't be trying to diagnose your player over the internet, so let's stick with the facts as you've laid them out: a player is disruptive, attributes much/most/all of that to ADHD, and you want to try to help them play without being disruptive. None of that is up for debate.
First thing to think about are the specific strengths and weaknesses this brings to your player. (In my experience they're one and the same.) The flip-side of inattention is the ability to quickly task-switch. The flip-side of fidgeting is working tirelessly at a trivial mechanical task. The flip-side of outbursts is spontaneous creativity.
With those in mind it's time to think about how an AL D&D game works, how it interfaces with those, how (much) to highlight areas where D&D plays to a strength, and how to mitigate those situations where D&D plays to a weakness.
In no particular order, here are suggestions for how to turn some of those things to your advantage, how to ameliorate others, and how to make use of your player's strengths. All borne from personal experience.
Use minis/markers/scrap-paper chits. Even if you'll never sketch out a map, have them out on the table to represent marching order, chatting with NPCs, exploring town. I keep a miniature croupier's stick in my game-bag and this, along with minis on the table that need to be moved every ten minutes, probably occupies ~30% of my fidget-time. This requires some self-control, as they can't be constantly playing with the minis in the middle of the table. But even during social scenes it can help to be all "So, Princess Vespa. At last I have you in my clutches."
Ask them to take good notes. It's a second set of ears to catch a detail you made up on the fly. It's a virtually-constant mechanical task if they embrace it. Flipping back through notes to find a piece of information is another side-task they can call on when needed. And you've got copious campaign notes. Where's the downside? This captures ~60% of my fidgetry.
Encourage them to sketch/doodle. The entrance to the temple you just painstakingly described. The layout of the buildings you mentioned in this village. Their character's signature spell going off. This occupies the last !10% of my fidgetry.
At this point, btw, we should also have solved the "I have to tell him the scene four times" and "he's on the phone" problems. He's likely on the phone because passively listening to something cannot occupy his attention. But "listen and jot the name and one-line description of every NPC, occasionally flipping to a little-used spell's description" may just be the sweet spot for attention and engagement.
Choose (your seat) wisely. You're in a super-distracting environment. Pick a corner/side table if you can. It may be tempting to seat this player with their back to the room--resist that. Seat them by the wall, facing everything. IME the visual distractions aren't the problem as much as the conversations at other tables. It's much easier to avert your eyes than it is to avert your ears. This, of course, helps your other players, too. At an AL location with more than two tables or multiple events going on, everyone's working overtime to filter stimuli. This also probably requires someone--likely you--to be good abut getting there on the early side. That's your talk-with-the-tough-player time: ask them to also come on the early side.
Schedule breaks. I don't know about your AL, but my sessions often run three-plus hours. I have a player at one site who really cannot sit for more than twenty minutes--your player may be in the same boat. (By the way, it's probably undesirable for all the other players, too.) Announcing and maintaining a practice of drink-break 1.5 hrs in, bio-break 2.5 hrs in can do a lot. If nothing else, it means that during those two times the player's up-and-about they're not missing new activity. But it's not nothing: knowing that there's a scheduled break at minute 90 can make it much easier not to jump out of one's seat during minutes 70-110.
Discuss length of scenes, beforehand if possible. "We're heading into a town you've never seen. Do you want to RP the getting-to-know-you-parts for the next half hour, or do you want to approach it some other way?" When your table's decided, you follow that decision. This way your player(s) who might have trouble focusing during those scenes have advance warning and they can bring their attention-management resources to bear.
Get in a habit of declaring intent when heading into a scene. A scene where the table's said "we're trying to buy access to the royal ball" does not sound like a good fit for some random-seeming blurted out idea coming from left field. But the scene "let's head to the docks and see what adventure we can roust" is a nice fit for a wild cannon. Get your table into the habit of declaring intent ahead of time, and thus let this (and all) players know when which playstyles are going to be appropriate or not. P.S. knowing your players' intent makes things much easier for you.
Final thoughts: