First of all I think they know that tantrums are not a good thing, otherwise they would do them in front of Sunday school class.
They do this in your home, because you are responding to the tantrums. From now on No whining is acceptable for any reason for anyone in the house and NO Tantrums ever.
When there is a tantrum out in public, leave. Get in the car and go home. Let you son know that his behavior is not acceptable.And his behavior is the reason you all had to leave. Do not go back even if he begs and promises.. You may need to do this a couple of times, but he is going to learn very quickly you are serious. If he does this at home, step over him and ignore him.. If he follows you go to your own room and shut your door or lock your self in the bathroom. Even if he begs and promises.
The rule is never give anything to the child that they asked for if it was during a tantrum or caused a tantrum. Once he calms down. Give him a hug and tell him in the future he needs to use his words. Tantrums are not acceptable.
This is not criticism it is an observation. "I decided to homeschool our oldest this year, but seriously considering putting him in school in sept. As much as I would hate for him to be away for that long," ~ is this a good reason to home school? This sounds like what you need not him.
I have a cousin that home schools because they live way out in the country and the bus rides each day would take a long time. They also have 6 children so they would end up in 3 different schools. I have friends that do not believe in some curriculum's taught in schools and want their religious beliefs taught on a daily basis and cannot afford private school, so they homeschool. But you not wanting them to be away from you to attend school, sounds a little clingy. Maybe your son picks up on this anxiety from you.
Let your sons know they are strong in their minds. Let them know they do make good decisions each time you see it. Also if they make a not so great choice, ask them "what did you just learn?" "How can you make that better next time?" If they feel insecure, let them know this is normal, we all feel this. If they feel like they do not want to share, they have that right, but they need to remember to respond with manners and to remember to honor others when they do not want to share. .
Also let them know they are smart enough they do not need to whine or to throw tantrums, because they know the words to use..
I am sending you clarity and peace.