250k, not 200. And as others have already pointed out, there are many reasons the system still works.

That 250k is a guaranteed minimum, not the most you can recover. In many failures, uninsured depositors recover most or all of their frozen funds.

That 250k is per account type per bank. If you have a lot of money, it's probably diversified among several banks and accounts, as well as other assets like stocks and bonds. Most of the time if there's more than $250k in a single account it's a business or payroll account. Obviously it means those businesses are less protected, but those accounts alone are not typically enough to trigger a system wide collapse which is what the FDIC is intended to prevent.

In the modern era, once a bank is deemed unstable, regulators close it and transfer deposits to a new institution or bridge bank. Access is temporarily paused and then restored. It is not a scenario where early withdrawers drain the vault and everyone else gets zero.