For me, I will still use the services of bank, they are good when you need them like for saving money, just save your money with them if you don't want to use it. Also they are good with lending once you built up a good relationship with them, you won't be having hard time to ask them money if you need so.
A bank's purpose as of this present moment is just really about saving your money and not gaining anything from it because we all know that the interest right now is the lowest of the low. Also people complaining about the fees on using services of the bank, I think it is better and cheaper compared to buying your own Personal Security Safe which can cost you thousands of dollars ( the security/protection you are getting in a banks is better compared to keeping your money at your home). Also some banks offer collateral if a bankruptcy ever happened which is a nice feature that protects your money being lost and also we can take advantage of creating several bank accounts with different banks so that we can diversify and maximize the collateral that each bank is giving.
Yes if you're sole purpose for using bank's service is just to save, you will be safe as they are covered by the law but don't expect that you will be earning good amount of interest. They are not the best place for interest or earning with your savings but they've got option for investments and there is a risk, it's up to you what you want to choose.