I don’t know enough about the big banks to make a comment or referral on those

Our “main” bank is Regions but I just looked and they are not good. They are in the southeast. We have free online banking and bill pay, images of checks, etc. All that you are looking for but I am not sure what “remote deposits” are so I don’t know if they do that.
We chose to park roughly 1/2 of our FFEF there because of hurricane evacuation. Almost anywhere we would go in the southeast has a Regions or in a nearby parish(county)/town.
Our other bank is a local bank. It is slower to implement some things but we do have free online banking through them.
These 2 banks do pretty much all we need. We have not had any purchases or transactions cancelled or rejected as you have. I am sorry to hear about your experiences.