As a landlord myself, I basically look for evictions, judgements, and general financial health. That can include having the same job for the past few years, and other signs of stability.
I would usually run the credit check myself, because some of those free reports don’t show evictions and court judgements.
The thing they tell you as a landlord, you have to do your due diligence. Some landlords are better judges of characters than others but at the end of it, a landlord is entrusting a renter with a valuable piece of property and it helps to at least do your homework.
I’m willing to wager that those landlords that you’ve dealt with that were willing to look at you and rent to you without a whole lot of documentation haven’t been screwed over by a tenant often enough.