Unfortunately I learned from another landlord on Zillow's Q & A before it was shut down that tenants (renters) abuse the emotional support dog certificate - either getting a doctor to sign it or fraudulently claiming they are entitled to an emotional support dog when they are not. This is part of the reason for the difficulty.
Another issue is homeowners insurance for landlords. Standard insurance policies do not cover liability or damage from tenants pets. Since landlords do not want to be sued, it's easier to either have a "no pets" policy or to accept small pets by weight. There are specific breeds which insurance companies do not cover period. Pit bulls are one of those breeds.
If you are renting, it is your responsibility to buy pet insurance with liability and damage coverage. Once you show proof of insurance, landlords will be more willing to accept you. The issue is if anything happens because of your pet, you will be sued - not your landlord.