It would? I don't quite follow the question. Yes, the second type of loan would invariably have a higher interest rate. Let's say there's two loans for 10000$ and that, regardless of the loan type there is a 0.1% chance that a debtor will have an accident. If the debtor is poor, they will be forced to choose between not making loan payments and (for example) losing a leg to gangrene. If the debtor is rich, they will can pay both their loan payments and medical bills at the same time.

Loan A asks for 1000$ in total interest and has enforced payment which will prevent a poor debtor from paying their medical bills. Regardless, the creditor has priority in payment and receives their interest either way. Expected value to the bank: 1000$.

Loan B asks for 1010$ in total interest, but has a hardship forgiveness clause. There is a 0.1% chance that the creditor will lose 10000$, but no chance of the debtor losing a leg. Expected value to the bank: 1000$.

The bank is indifferent between the two loans, as both have the same expected return (lets ignore variance for now on the bank's part; we can assume they deal in a large enough volume of loans or charge slightly more interest to compensate). The poor debtor prefers Loan B, as 10$ is a small price to pay for protection against crippling disability. The rich debtor prefers Loan A, as they do not want to pay 10$ to avoid a negative result which they are already protected against.

I don't see any particular social problems arising from this situation. Do you?