Larger cats have been tamed, and used for hunting.

In particular, the cheetah and caracal were used for hunting, with the cheetah used in Egypt, the Middle East, South Asia, Central Asia, and China, and the caracal in South Asia and China. (Chinese examples were imported.) Each of these cats has strengths that make them attractive as hunting animals: the cheetah is famous for its high running speed, and the caracal is a superb jumper, able to jump over 3m, and can catch birds on the wing (as long as they aren't flying much higher than 3m). Cheetahs are daytime hunters, and while the caracal is primarily nocturnal, they will hunt during the day. This makes them much more convenient hunting animals than cats which will only hunt at night.

However, neither the cheetah nor the caracal were domesticated, despite being in such wide use as hunting animals that there were almost certainly repeated attempts to domesticate them. The cheetah is notoriously difficult to breed in captivity, with only one successful breeding recorded before 1956. If you can't breed them in captivity, and have to capture them from the wild, you will have tamed cheetahs and will not develop domesticated cheetahs. Caracals don't appear to be as difficult as cheetahs to breed, but they are usually solitary animals, temporarily getting together as breeding pairs but otherwise living alone. Their solitary nature will make them harder to breed in captivity.

Difficulty in breeding isn't the only thing that might make domestication of various cat species difficult. The large cats (e.g., lions, tigers, leopards) can be very dangerous to humans, and such dangerous cats are relatively rarely tamed (cheetahs pose little danger to humans, unlike many other large cats). Cats which mark their territories with stinky urine (and don't stop doing so even if neutered) are unattractive as house pets, which reduces demand. Cats are also fussy about their diet, unlike dogs which can be raised on almost-vegetarian diets. This makes them less efficient as hunting animals, when you consider what they catch vs what they eat. (Dogs are much more efficient in terms of the amount of meat required in their diet, and have been raised in various places as sources of meat for human consumption.) As already noted, daytime hunting is also an advantage. Social behaviour, such as a hierarchical organisation where a human can play the role of the dominant animal, cooperative hunting, and more, can also make them more attractive as pets, hunting animals, and working animals.

Together, these problems explain much of the lack of domestication of many cat species. Being any of: dangerous, nocturnal, solitary, stinky urinators can make them much less likely to be domesticated. If there are other suitable hunting, work, and/or companion animals already domesticated, why try to domesticate a new species which could easily be more trouble than it's worth. Noting that the dog was domesticated before civilisation, and the domestic dog spread across Eurasia, Africa, and the Americas, domestic dogs were readily available in much of the world.

Two other cat species should be mentioned: the serval which was kept in Ancient Egypt (perhaps as wild-caught tamed animals rather than domesticated) and the leopard cat (Prionailurus bengalensis) which appears to have been (at east partly) domesticated in Neolithic China. Both appear to have been replaced by the domestic cat.

The domestic cat violates some of the things that make the "perfect" domesticate. They hunt alone, often nocturnally. They are not meat-efficient in terms of what they're hunting brings in. They don't have the strongly hierarchical society of dogs. However, their main value as hunting/working animals is rodent-killing. This is useful to farmers (and anybody who stores grain and other foods that will attract mice etc.), even if they have no intention of eating the mice that their cats catch. Indeed, mouse-hunting is likely something that brought humans and cats together in the first place - as human agriculture and food storage attract mice, the mice attract cats. Many other cat species, such as lynxes which don't typically hunt mice, don't have this advantage when it comes to domestication.

In summary, multiple cat species have been tamed, and some of them domesticated. However, the spread of the domestic cat appears to have replaced other domesticated cat species such as the leopard cat (perhaps because they were less stinky urinators). Other cats were esteemed as hunting cats (the cheetah and the caracal) but were not domesticated - needing to be wild-caught and tamed, they were hunting animals for the upper classes, rather than for common folk.