You are officially off my cat’s Christmas card list!
Cats are not as asocial as many believe. They get along fine in groups and I have no doubt if competition requires it, they would evolve from effective ambush hunters into effective pack hunters like their lion cousins.
Cats vs mustelids? Mustelids have a very high metabolic rate, which helps make them skillful hunters, but this comes at a steep resource price. They need to eat much more than cats (~7 x more). Who would survive best in an environment with limited food?
Although cats require less food than weasels (a mustelid) to survive, one-on-one the cat will usually kill the weasel in a fight to the death. The weasel is skillful at farting, but that isn’t a match for 5 sharp daggers on the distal phalanges of each cat paw.
House-cats have not lost the ability to survive in the wild. There is little difference between them and wildcats. They are outliers with regard to domestication. Surviving in homes and blighted urban areas are just additional skill-sets they’ve mastered. They adapt.
Certainly many cats would be killed by larger predators (at least until they evolve), but they can make up for the losses. Cat experts calculate that one female cat and her offspring will produce 100 cats in seven years.
When house-cats join the stray and feral cats I believe they will survive and thrive in the wilderness. Increased predation and competition for resources will spur them to evolve into even more formidable alpha predators—bigger, badder, and smarter.
Today Snowball and Tigger play with toy mice, honing their skills. Tomorrow they will be impaling wolves, coyotes, bears, mustelids, and octopuses with crossbow arrows (or maybe retractable ballistic claws).
I say this soberly with confidence and solemnity: pussycats will conquer the world. 
Warning: they may do so even before humans go extinct, so hide your kids and puppies.