If he's complaining and nagging, that's interesting to me - because he's not actually afraid, it seems. He doesn't get agitated or sit in a corner, he doesn't shake or cry, he's just complaining. That says to me it's not anxiety.
Now, why he doesn't want to play with the team is another story. He willingly went to the practices, wanted to join, etc. So that doesn't sound like anxiety. He didn't want to work as hard as the coach expected, or he found that the other kids were really good and he felt he didn't measure up. That's either fear of failure (which is different from anxiety) or a lack of desire to work at the higher level of this more competitive team. Sometimes kids who are "the best" at their sport (or theater or debate or karate….) get freaked out when they get in with other kids who are just as good or better.
He's also fighting you on the therapist. Unless there is a major stumbling block or there was an episode, it sounds like he doesn't want to work at that either.
I would absolutely talk to the therapist, but if this were a kid without anxiety or any other psychologist issue, I'd say that you have extremely limited screen time (games, computer, TV) and a requirement of X amount of time outside (and type of physical activity) or playing something creative (Legos, art, anything) and reading (doesn't matter what - just reading). At 11, he's too big for you to carry out the door, so all you can do is take away the stuff and tell him he sits on his bed until he does one of the acceptable activities for the designated amount of time. If he complains, the clock goes back to zero. He will only be able to do that for a half hour and then he'll cave.
You HAVE to be emotionally ready to be the parent here, not his friend. I know it's hard, and it does seem easier to give in. But that's doing it because it's for YOU, not what's best for HIM. You will create a situation where he is ruling the roost. He will learn that complaining is an effective strategy. That will backfire immensely and you will have an even worse situation when he is 13 and 15 and 17. You have to turn a deaf ear to it, tell him you have heard his wishes and you aren't agreeing to them, and no amount of complaining will change your mind. In fact, it will make things worse. For example, let's say you allow 2 hours of screen time a day in the summer. Every time he nags, you take away a half hour. 4 bouts of complaining, and he's done for the day with zero time. If he keeps it up anyway, start taking away from tomorrow's allowance too - or the whole week. If he gives you the "I'm bored" speech, have a list available of stuff he can do - laundry, weed pulling, lawn mowing, toilet cleaning, floor washing, etc. Stuff that has to get done anyway so he might as well since he is bored and you are busy. Be willing to walk away from the argument - don't engage once you each have stated your position.
But since he has a therapist, I'd discuss this with her/him first and make a plan you can stick to. The child is not in charge though - he does not get to decide what he does all day. Don't forget that you are building a man here! "I am the parent" - that is your mantra!