Earlier quoted context omitted.
What do you mean by "It is $5/mo for accounts with less than $1000, but you're not going to get better than that at a bank."? Aren't there lots of banks that charge nothing even if you only have <$1000?
There aren't many anymore. Banks mostly eliminated or heavily restricted free checking accounts when overdraft fees got capped. You may be able to find a couple credit unions that do this, but you still have minimum account balances and other hurdles that make it difficult to open an account if you have bad credit and little money (not to mention time or financial savvy).
I don't know about the credit check part; I set up with my parents years ago so that may have exempted me as I didn't have any credit history at the time.