I'm going through something similar myself. I'm trying to find a place to life in Miami, and I have 3 dogs (< 20 pounds each), and 8 (!!!) cats. Nobody wants to rent to me, even though I'm looking to spend a *lot* of money on a place. The close-mindedness of HOAs really surprises me -- HOAs call themselves pet-friendly, but allow 1 pet less than 20 pounds. It's ridiculous.
However, I'm pretty sure that's a business opportunity for more open-minded people. So I'm going to start buying places outside of such HOAs, and I'm pretty sure I can charge a premium to be an actually pet-friendly rental. It'll take some creativity to avoid legal and financial challenges -- things like requiring a weekly cleaning service that'd report to me, the property owner, to alert about ongoing damage. Still, affluent pet-owners (like me, currently) would certainly be willing to pay a bit more for such freedoms.
I'm most concerned about legal risk from dog attacks and such, but I'm pretty sure that could be overcome with good insurance and contractual indemnification.