It's nice to see someone taking steps to verify and check a prospective tenant's background. I've met other landlords who just do a basic credit check and hand over the keys once they receive the deposit and first month's rent, then get surprised when they end up in court. The current landlord's recommendation should be taken with a grain of salt. It's better to call one of their previous landlords--they won't have much to lose by giving an honest recommendation. If this prospect checks all your boxes (real proof of income, good rental history you can verify), then it would be fine to rent to them.
If you ask me, the better question to ask is: "What are your criteria for qualifying tenants?" This can indicate how much trust you place in your processes. If you have a low threshold for qualifying tenants, then I can see why you are uncomfortable making a decision based on just those qualifications. If your criteria require a potential tenant's credit score is over 700, income is greater than 3x the monthly rent, and their rental history shows stability, then trust your process. Otherwise, what's the point of having processes in place?
Best of luck