Here's a variation on your theme. Suppose David is a potential client of someone like Jane, but can't find anyone who is in the Jane business, because each Jane can only handle a couple of clients.
David finds a homeless person Frank, and proposes a business arrangement. He gets Frank some decent clothes, arranges for him to get cleaned up a bit, maybe pays for a haircut and shave. If Frank does not currently have the documents needed to open a bank account, David helps him to obtain these legally in the usual manner. If Frank does not currently have a valid residential address, David makes appropriate arrangements, e.g., perhaps "sharing" David's place (not really though, just receiving Frank's mail).
Once he is presentable, Frank goes into the bank and opens a non-interest bearing account with David as the DBA name. Later he gets an ATM debit card for the account, and activates Internet banking.
Frank gives David the debit card, the PIN, the password for Internet banking, and a bunch of deposit slips. Thereafter, when David wants to cash a check, he needs to find Frank, for him to endorse the check. Then David deposits the check, and starting 10 days later, withdraws a few hundred dollars a day in cash from the ATM.
It's unclear how much Frank would charge for this service, but there are lots of homeless people, so David can shop around a bit. David will be spending some up front money for the clothes, haircut and so on, plus any fees for getting the ID documents. The main risk is that David will be unable to find Frank when he really needs him, or that Frank will suddenly get greedy and charge a lot for the endorsement.
Also, unlike your original scenario, the checks are actually deposited, not just cashed, so some paper trail is left. But as Eric pointed out, the bank may "virtually" deposit the check anyway. And the account is non-interest bearing, so as long as the deposits are not too big or too frequent, it shouldn't raise any flags.