Actually, people have tried for a very long time to domesticate animals like lions, and it doesn't work the same way; some animals are much easier to domesticate than others. Generally speaking, you need:
- A pack/herd animal that conforms to a social hierarchy.
- Eats whatever's around/isn't expensive to feed.
- Breeds in captivity.
- Are docile/don't blindly panic and flee when startled.
- Reach maturity quickly relative to the human lifespan.
- A society motivated and dedicated to domesticating this animal.
I believe the only currently domesticated animal that breaks this is the cat (breaks 1), and the domestication of the cat was a much longer process than, say, the cow and happened in a very different way. There's also a lot of evidence to suggest that dogs and cats basically domesticated themselves into our societies because the ones that rooted through our trash in a non-threatening way tended to survive better.
For this reason, it's very unlikely we could domesticate things like bears (no social hierarchy).
Lions have social hierarchies, but are a lot more aggressive and dangerous than wolves, so it's difficult to capture enough of them at a pre-industrial technology to even begin domesticating them. They're also more likely to view humans as competitors, not potential partners, like small cats and dogs did. Finally, they're somewhat difficult to breed in captivity and can't stand small enclosures.
An example, the Americas have all manner of animals, but the Native Americans were only able to domesticate: guinea pigs, turkeys, chickens, dogs, alpacas, and llamas. Bears, buffalo, large cats, birds of prey, deer, and snakes might seem like things that would be advantageous to domesticate, but in reality it costs far too much, and most of it upfront. You're likely to get killed off by a neighboring tribe that isn't routinely wounded and wasting thousands of pounds of meat on unruly carnivores generations before your payoff occurs. Or, even, killed by your new "pets"; bears and lions have a ranging territory of dozens of square miles, so it's hard to keep them happy in a cage.
A lot of my information comes from the fantastic book Guns, Germs, and Steel by Jared Diamond.
Edit: Your clarification edit makes my "answer" more of a rebuttal of your premise than an answer to your question; I'll leave it available for future readers.