From my understanding, there aren't really any good alternatives. Feline stomatitus is the overreaction of the immune system to the various bacteria, plaque, and tartar that naturally accumulates in the mouth, especially around the teeth. So the known treatments are related to somehow reducing the amount of bacteria in the mouth.
Medications such as antibiotics can temporarily reduce the symptoms, but feline stomatitus is a lifelong illness. Not only will these medications have potential side effects, which also can be unpleasant for the animal, they will not cure the illness. The bacteria in the mouth will become resistant, and the cat will end up in the same position as before.
The other option, keeping the teeth consistently clean, is not realistic for most cats, as they usually won't allow you to thoroughly brush all their teeth, especially when stomatitus is painful. Perhaps you might be able to increase your habits to keep your cat's teeth clean enough that you could see improvement, but realistically it would be extremely difficult if not impossible to brush any cat's teeth enough for the stomatitus to no longer be an issue, when pretty much any cat will at minimum not be thrilled about it.
Meanwhile, removing the teeth has been shown to have much more likely than not a permanent and quite positive effect on the condition.
On the positive side, cats do very well even with no teeth at all, as long as they are kept indoors. Since cats are carnivores, they don't have much need to chew, and even with teeth, they don't normally chew all that much. Their teeth are more for hunting, self defense, and tearing off chunks of food small enough for them to swallow. Wet food is easily eaten without teeth, and some cats can even manage dry food without teeth, although surely wet food would be more recommended.