There are more scenarios that a landlord or property manager can see in a lifetime than could ever be noted. If you're in the business for any length of time you'll see good tenants, bad tenants, bad tenants that you thought were good in the beginning, and tenants that started off a little rocky but evened out in the end. While each scenario is different, that doesn't mean that your process should change for each person. Tenant screenings are a necessity, even if you can't possibly imagine a nicer person in your property.
Imagine this scenario: A young couple comes to you and wishes to rent your property. They're well dressed, well spoken, and have excellent references. The only problem is that they say they've recently been the victims of identity theft (it's happening more and more, so this is entirely plausible) and they've put a freeze on their credit report. They would really appreciate you accepting the references. They're even willing to pay a larger deposit. Great, right? Well, they may be honest, hardworking people, but what if you'd run their credit to find that they weren't so honest with you? You could have accepted people into your home that never pay on time, that have a habit of leaving destruction in their wake, or even tenants that are willing to cook drugs in your rental home, leaving you to clean up and pay the rather large price-tag attached to that. Needless to say, the old expression about not judging a book by its cover can certainly relate the screening your tenants. Following up on references is a great practice, but don't limit yourself to that. If a person is not willing to have their credit and criminal background run, they may have legitimate reasons, or they could be trying to hide something. Are you willing to risk it on your property?
If you want to read further on how tenant screenings can help your profitability as a landlord, check out this article from BiggerPockets here.