I disagree. There is a huge difference between correctly caring for a big cat and correctly caring for a snake. The difference is that big cats cannot thrive in a house setting. Many rescue organizations and zoos have done wonderfully at allowing big cats to thrive in captivity. The difference here is that rescue organizations and zoos (at least the good ones) do not view big cats as pets, and they (again, the good ones) minimize human contact with the cats.
The truth is, while the love and care we put into housing and maintaining our snakes may be sufficient for snakes, big cats require much, MUCH more. For one thing, snakes are inactive animals in the wild and can do quite well in appropriately-sized cages. Big cats (in my very firm opinion) should never be kept in a cage. If you examine the size of the territories they establish in the wild, that is simply not something an individual can provide. Not to mention the cruelty of capturing and importing tigers and the like.
"I believe we all have to the right to own animals as long as we take care of them correctly"
I agree with this. I just don't think it is possible to do with big cats.