A typical bank branch sees a few people per day. I could deposit thousands of checks a day. It would be physically impossible for the bank to handle. They could implement a policy to handle this but then they are denying that their branch is able to handle it. Is it possible to simply destroy the banking system by depositing large numbers of checks? Is there any legal consequence?
You greatly underestimate the capacity of banks to handle high volumes of transactions. There are many corporate and governmental bank customers that deposit thousands of checks a day. At the peak of tax season, the IRS deposits millions of them per day.
Nationwide, about 4,389,000,000 checks are deposited each year and 17,500,000 checks are deposited each day (in the entire U.S.).
You would need millions of checks per day to overwhelm a bank. And, while it is not a crime to deposit a check, it is generally a crime to deposit a check written on a cancelled bank account or with insufficient funds. So, unless you have a lot of money to deposit (which people who try crazy schemes like this one usually do not), sooner or later, you will run out and you will start knowingly writing bad checks, which usually is a crime.
Banks are free to decide that they will not do business with you as a customer, or will establish special procedures for dealing with you in particular (e.g. assigning you a "personal banker"). They are also free to refuse to, for example, handle deposits of checks under $1, and can enact policies limiting the number of deposits a customer can make per day (most mobile deposit systems already have such limitations). Many bank accounts also charge a fee for each check deposited, so if you deposit 1,000,000 checks per business day, you are going to be paying the bank an immense amount of transaction fees. And, most deposit agreements may be amended prospectively by the bank at any time with notice to you if they find that they want to change its terms.
Simply put, as a practical matter, your evil plan to deposit the bank to death won't work. There is no viable way to conduct a sustained "denial of service attack" on a bank by making too many deposits.
Have you ever actually deposited multiple cheques at the same time?
People hardly ever use cheques anymore (except perhaps in the US), but a few years back when I used to deposit about 25 cheques at a time, the teller would simply stack them into a machine that fed and scanned them all in a second or two. The teller didn't even bother looking at them.
I could have just as easily given them several hundred cheques and it wouldn't have bothered them at all.
How long would it have taken you to write all those cheques though?
You can't "destroy" a bank by demanding account services. Banks in the US generally include in their agreement with depositors that the bank has the right to close the account at any time for any reason, and if your demands are onerous, they'll simply exercise that right.
I think what will happen in real life is after a couple hundred checks, they'll ask you to upgrade to a business account, or some similar higher tier, for which there may be a higher fee that covers their costs to service your account. If you decline to do so, and keep bringing in the checks anyway, you can expect they'll cut you a cashier's check for your account balance, say "Sorry, you no longer have an account with us", and tell you to get the heck out of their lobby.
Most bank branches see hundreds of people in a day, large branches see thousands (or did pre-COVID), There are automated scanners for depositing checks, depositing thousands would in no way overtax a single branch, and if it did, the checks could simply be sent to the central HQ where tens of thousands of check can easily be handled.
It is not nearly that easy to "destroy" even a single branch, much less the overall banking system.
But if a particular account holder became a problem for a bank, it could and would simply terminate that person's account, as Nate Eldredgepoints out in a comment.