The mortgage contingency is the sweet spot for you to find your renter. You committed to buying the home, your mortgage process is underway, and the contingency clause offers a little protection - in case you can't find a renter in time. If you have trouble finding a tenant, you can still back out of the deal, and the seller will refund your earnest money.
And while it's true that 30 or so days seems like a strict deadline to meet, again—the real estate and rental markets move fast! During the contingency period, you'll have time to advertise and show your property to prospective tenants, then draw up a lease agreement for your renter(s) while your mortgage lender gets everything together to process and approve your loan.
If you don't qualify for both mortgages outright, you'll need to provide your lender with proof that you're renting out your current home before the contingency period expires.
Typically, the lender wants copies of the lease and the security deposit check. The check proves that you have a renter and the amount of the rent. We will process and approve the loan with these documents, and you'll be ready to close on your new home.