They would miss 2 practices and the game. That's not too much. I would do this, yes, as the beginning of your new training session towards better behavior.
Drawing a line in the sand is a good thing. When you do, you MUST follow through no matter how much he whines, cries and protests.
At 10, he's old enough to understand consequences. The 7 year old is a lot younger developmentally and emotionally, and he is monkey-see/monkey-do where his brother is concerned. He feeds off his older brother's behavior. What you have to do when they are acting out together is separate them. With summer coming up, you need to put a plan in place and stick to your guns and be willing to make them miserable (and yourself some too) in order to drive the lesson home.
When you are going somewhere, once they are strapped in to the car, before you turn on the ignition, face them and tell them point blank what you expect of them. Tell them exactly what the plan is, where you are going, why, what the trip is for, what it is NOT for... if it's a looking day and not a shopping day, make sure they know that you aren't buying anything. And stick to that. Right now you cannot afford to be "flexible" because they see it as a sign of "give an inch, take a mile". Tell them that if they misbehave, you will give them 2 reminders - 2 strikes - and then the 3rd strike means that they get back in the car and go home. At home, they are in separate rooms for at least one hour and then they have to do some chore they don't like as an apology to you for all the trouble they have caused you.
You must be willing to leave a cartful of groceries in order to get your point across. Turn the car around and don't see the matinee. Leave the restaurant and stand outside to get it bagged up. March them to the car within 5 minutes of getting to the park or library or ANYWHERE. If you know specific places that it's worse, like skating, putt-putt, bowling, then you sit down with them and say that you're going to try it with them, but if they misbehave, you will be leaving at the 3rd strike and it will be a long time before you go again.
I had to do this with my older son who was a difficult personality and would misbehave. It took months to get him to see that I meant what I said and that I would continue to curtail his fun until he straightened himself out. It was so hard on my younger son who was not a problem. But it had to be done. When he realized he lost the privilege of the outing, had to stay in his room without his brother for an hour, NO TV, NO gameboy, NO electronics, NO nothing but reading, and he had to do a chore he didn't like, his behavior gradually got better. He was impulsive and bull-headed and had trouble thinking ahead and governing his emotions, but I consistently gave him the same consequences and it helped him gradually think before jumping off the cliff, so to speak.
It will get worse before it gets better. He will punish you in order to get you to stop. Don't let him do it. Be strong. You'll be glad you did when it comes to 5th grade...