Foundation Savannahs (typically the first three generations bred from the serval, but in reality anything that is not SBT) are considered hybrids by TICA, SBT Savannahs (those Savannahs with three generations or more with only Savannahs in the pedigree, so no outcrosses or serval blood) are considered purebreds and a domestic breed. USDA considers any cross between an exotic and a domestic animal to be domestic. Every state has their own interpretation of what our Savannahs are - so as you can see, there is no good answer to that question as it depends on who you ask.
Savannahs are a domestic breed that descends from the serval - not the jungle cat, or the Asian leopard, or the caracal, or the Geoffroy, etc. Mixing other wild genes only muddies the water and detracts from what the breed is trying to achieve - a unique looking cat that is different from every other breed of cat. If we mix in wild genes from Chausie ancestors, Bengal ancestors, Caracat ancestors, or Safari ancestors, etc. we can never accomplish that because each of these breed have their own unique look.
The point of creating a Savannah was - and is - definitely NOT to create a larger breed. There is nothing in the Savannah standard that says it should be a large breed, and in fact, the vast majority of later generation Savannahs (e.g., f3 and later) are the size of any other average to large domestic cat. We DO want to create an exotic looking cat, but again we want it to resemble the serval, not the jungle cat, Asian leopard, caracal, or Geoffroy cat.
If someone wants to mix wild genes to create larger cats that is their prerogative. I don't think it's a good or bad thing, it's simply someone's goal for their breeding program. However, I frown on them taking advantage of registering these cats as Savannahs (or whichever breed) because they are too lazy to actually develop their own unique breed, and then riding on the marketing strategies of Savannah breeders (or of whichever breed) to advertise and sell their cats for big bucks.
I'm not sure who said that if the cats are not TICA registered they are not a legitimate breeder. You don't need to have a TICA registered cattery to register litters and kittens as long as one of the parents is also TICA registered. I think more to the point, if the breeder and cats are not registered then how do you know what you are paying for is really what is advertised, be it a Savannah or other breed? The cats could be moggies off the street that the owner decided to breed and because they have spots call them Savannahs and sell them for thousands of dollars...
I also don't know who said that TICA is 'only' a registry. TICA is definitely a registry, a genetic registry, so it allows us to register and track our breed. In addition, it hosts hundreds of shows every year, which encourages breeders to work even harder at developing the best possible Savannahs through the competition of showing. In addition TICA promotes camaraderie and community amongst breeders, and encourages conscientious and ethical breeding habits.