If you want better service, there are two ways. One is to be an ultra high net worth individual. But the threshold for that is pretty high - from what I recall, real private banking nowadays starts around $5M to 10M in assets with a single institution. The term "millionaire" doesn't mean as much as it did in the 1990s.
The other option is to go with a small local bank or a credit union. If you're in a CU with 10,000 members, it's a lot easier to resolve problems or discuss unique circumstances.