Obviously a person who keeps tigers should require permits, insurance, and some form of oversight--these are animals which might escape and, well, eat people.
However, the same cannot be said for ocelots. No one's ever been eaten by an escaped ocelot. With basic guidelines for how such animals should be kept, why aren't existing anti-cruelty laws good enough?
There's a movement in this country now to restrict or ban pet ownership, starting with the most vulnerable fringes. The phrase you used is a symptom of it--'some animals are wild and should not be kept as pets'. What does that mean? If an individual keeps an animal, does that mean that it is a pet...that they will pet it? What if the care they provide is superior to that provided by a zoo? What if they're propagating an endangered species in captivity, because it has virtually no habitat left, and there's no where else for it to exist?
Domestication is a process by which an animal evolves to become more suited to live with humans. I won't say that it's a process that we deliberately guide and control, because there's some evidence to suggest that the domestication of the dog was inadvertent, not deliberate. What we mean by 'more suited to live with us' changes between species, though. The koi fish is a domesticated species, yet it hasn't been bred for additional docility. Why? Because that wasn't necessary. Wild carp are docile enough already. Gerbils are a commonly kept pet--and have not been significantly altered from their wild form. They are not a domesticated species, because changing them wasn't needed to make them good pets. That's an important point to keep in mind.
Not all wild animals
make bad pets. Not all domesticated animals make GOOD pets. The Spanish Fighting Bull is unquestionably a domesticated animal, and it would as soon kill you as look at you. It's not about 'domesticated' or 'wild', and it never was. Those are words used as tools by people who have an agenda, and they use them skillfully enough that very few people even notice they are being manipulated. It sounds good, so it must be true, right?
Think about it carefully the next time you see a proponent of animal rights talking about wild animals and domesticated animals, and pets. Remember that animal rights people want to end pet ownership. They consider it to be slavery, and they consider domesticated species to be an abomination.
See how easily they were able to skew the facts about ferrets, to claim that they're bad pets...to convince people that they are 'wild' even though they are domesticated animals. The truth is, ferrets aren't for everyone--neither is any other pet. It takes a very special person to own and properly care for a lion. It doesn't take an entire zoo--just someone who's willing to do things the right way. Those people DO exist. Why shouldn't they be permitted to pursue happiness in their own fashion?
A lion, housed correctly, has a reasonably good life. There are some things it would do in the wild that it doesn't get to do, but there are also benefits that it deeply enjoys--it's a trade off. That lion never starves, never goes thirsty, and is kept free of parasites, and treated for any injuries or disease it suffers. It enjoys social interaction (either by being kept with another, or with its keeper), toys to keep it from being bored, and peace.
Animal rights folks would tell you that the lion would rather be free. That's no more accurate than saying that the lion would rather be captive. It's an opinion that has nothing to do with what the lion actually wants. If the lion doesn't show signs of excessive stress, and appears to be content, then it's doing fine--that is the only conclusion we can draw, unless you want to try to telepathically communicate with a lion.
It's becoming increasingly true that there isn't anywhere left on earth for some species to exist, other than in captivity. This is as true for smaller species as it is for larger ones. An entomologist recently rescued a unique species of roach that lived in bat caves, by propagating them in captivity, and then sending colonies to a bunch of folks involved in the insect keeping trade. The roach was rescued hastily, because the cave it lived in...the only place on earth that it lived wild...was slated for Bauxite mining. It is assumed that this cave is now gone, and so is every wild roach of this species. Yet, they live on...because people are keeping them as 'pets'.
Should they go extinct because 'wild animals don't make good pets'? We have some hard choices to make. I've seen a view expressed which I personally found very distasteful, and had trouble understanding. It was that a species 'out of context' was worth nothing--that it was not worth saving. That it was useless, and should go extinct. This person felt that if no habitat was left to support a species, it should disappear.
I believe every species has intrinsic value, and should be saved, even if it means we must keep it, and provide what it needs to continue to survive.
Steve Irwin pointed out that humans love what they can touch and interact with, and that they will only save what they love. A world without non-domesticated pets is a world in which there is no wild left at all, IMO. When animals become an inconvenience to humans, something that is just in the way, rather than something to love, then they quickly perish.
No one should ever have to name a new, tiny chameleon "Brookesia triste
' ("sad"), because it was found in a tiny patch of isolated forest next to an expanding city. That this little chameleon may go extinct so soon, leaving only a photo behind--that is the tragedy. A new roach "Simandoa conserfariam"
("live in many places"--because it exists only in captivity now), inspires more of a sense of wonder along with the sadness, doesn't it? It's not sad...it's alive. That means there is hope. Does the roach care that it no longer lives in a bat cave? It's found a new way to live. Isn't that what evolution is all about?
There's no way to leave things the way they were before we arrived on the scene. There are ways to help other species survive in spite of our presence. What is 'right' in all of this? Everyone will have to decide that for themselves, but be very careful before you tell someone else that their conclusion on the matter is the wrong one. Every course has consequences, both good and bad, and either way, something is lost.
So, do wild animals belong in the wild? Yes...if there is wild left for them. Do they not belong in people's homes or yards? That's a separate question--I believe that, sometimes, they belong there too.