Yeah, this is a terrible idea.
First, if your cat is sensitive, how will it handle living with total strangers for months at a time?
Even if it does handle this well, on those pet sitting sites, you know that the people are looking specifically for a pet-sit arrangement. The site allows people to review others and give references. It's a risk, but less of a risk than listing it as a rental and hoping that whomever wants to rent it is actually a decent cat care person.
It also just makes the whole process of finding good tenants that much harder and more complicated.
I agree, either rehome the cat, take it with you, or find a trusted friend/family member to take the cat for a few months at a time, but only if the cat can handle that, which it might not be able to.