With limited credit, a lender could potentially ask for a co-borrower - adding strength to the credit - not payment ability. Credit is necessary for a loan - one of the 5 C's of credit - character - the intent to repay..
There is nothing in the credit basics that reference maintaining a checking account. Other types of accounts (savings) are available too.
As far as underwriting without a credit report, it is possible, and many banks do it to get people started. Credit can take the form of rent, cell phone payments, payments to utilities (gas, water, phone), corner grocery store, etc. In inner cities, informal (not credit bureau) credit has a significant place. I believe that one of the secondary market buyers has a program for making loans without a credit bureau.
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Integrity. With it, nothing else matters. Without it, nothing else matters.