Co-signer? Not allowed.
This is the part which confuses me the most. My personal experience has never come across any sort of lender which wouldn't allow cosigners. Sure, some means-tested home loans won't allow cosigners that don't meet the same means-test as the primary borrower, but they still permit and actively want qualified cosigners. As a matter of underwriting, adding a cosigner always increases the likelihood of collecting on a defaulted note, reducing risk for the lender. There's zero financial sense for a lender to not allow a cosigner.
I hope you found a different used car lender, ideally one which offers something more reasonable than a 72 month (!) term.