Legality
They cannot refuse to accept coins and demand some other payment after providing a good or service. Legal tender is legal tender for all debts. But until they provide the good or service, they don't have to accept it.
In this case, you want the service of depositing money. But by its nature, they have to accept the payment first. In that situation, they can refuse it. There is no law that banks have to accept your deposits. If they don't want you as a customer, that's their problem.
Consider switching banks. Historically this was easier and some banks may still do things the old way. Call your local banks and ask. Perhaps you'll find someone happy to do business with you, on your terms.
Depositing coins
As already said, some coin rolling machines will pay you with gift certificates. If you plan to buy a sufficient amount from the place that accepts the gift certificate, this can get that place to play the fee. That may help you, although it is obviously a limited solution. The goal is to make it so that you only make purchases that you would have anyway. The seller obviously has a different goal.
It's possible to buy coin sorters. Heck, you could buy one with a gift certificate from a public machine. Cheap ones require extra work to get the coins rolled and may jam a lot. More expensive ones do more of the work for you. Note that a given sorter that works better may be cheaper than another that doesn't work as well. Cheap is more of a qualitative judgment than a financial measure in this case.
Spending coins
If you carry a small amount of change with you, pretty much everywhere accepts small amounts of change for purchases. So if you have been always paying with dollars and dumping the change in a jar, instead always give the correct change (coins). They may still give you dollars in change, but at least you won't get new coins. And you'll use some of your existing coins. Of course, this doesn't scale well.
For small purchases, say $1.50, you can often pay the whole thing in change without argument. Or if something is $18.50, you might give them $10, $5, two $1 bills, and the rest in change.
If you are buying something and can see that they have little change in one of the coin buckets, offer to swap some change for bills. Sometimes places find that easier than breaking a roll.
With vending machines, use change instead of dollar bills. Especially use exact change so as not to convert bills to change. They usually don't take pennies, but they're great with nickels and above.
This won't allow you to use change as a way to force yourself to save. But it will keep your change down to a manageable level going forward. And you might be able to use up your existing store. I'm assuming that this isn't a fifty year coin collection that you are just now starting to process. But if you have six months of change, you should be able to use it up in a year or so.
I tend to do this. So I rarely have more than a couple dollars in change. No one ever tells me that they don't take change, because I don't give anyone a lot. Maybe $.99 here but more likely $.43 there. Sometimes I give them, e.g., $.07 so as to get $.25 in change rather than $.18.
It's a little more work at every transaction, but it saves the big clump of work of rolling the coins. And you don't have to buy wrappers.
