To be honest as a Landlord, the problem much of the time is not the pet, but the owner.
And the landlord not knowing you or you standard of living. Sadly Not everyone is the same, and it is far easier to have a blanket NO PETS Policy.
From a landlords point of view, it isnt easy maintaining a property to a high standard, costs wise, and a good landlord will decorate and freshen up before each tenant moves in as a matter of course. maintenance costs money.
You are from what i have read the exception to the rule on the whole.
If 2 months after you moved in, i found you had a dog. Your contract would end without exception. A landlord might take a different view, if you had been an exceptional tenant for a fair period of time, then perhaps an exception could be given, based on your good tenancy value.
Sadly the Landlord has more tenancy choice than in recent years, so they dont have to make exceptions, if you break your contractual agreement, they will simply find another tenant.