More to the point, one of a number of reasons that the tuataras, the only remaining species of the order Rhynchocephalia, have survived in New Zealand is that they are not skinks, geckos, or more generally members of the order Squamata. The order Squamata includes lizards, snakes, and another related group of reptiles. The tuataras are the only reptiles in New Zealand that are not members of the order Squamata. As the only other terrestrial reptiles native to New Zealand are lizards, more specifically skinks and geckos, the tuatara has no close relatives in its environment, or actually anywhere in the world. Rhynchocephalia probably split from Squamata in the Permian, more than 252 million years ago. So there are no close relatives of the tuatara that can give the tuatara diseases.
Furthermore, the tuatara's competition, New Zealand's lizards, do have closely related species that can give them diseases. There are many species of skinks and geckos in New Zealand, all of them unique to New Zealand. So perhaps all the skinks evolved from an original New Zealand skink species, and all the New Zealand geckos evolved from a original New Zealand gecko species. This would make them closely related.
But even if that is not true, we can be sure that New Zealand geckos pass around gecko diseases and New Zealand skinks pass around skink diseases.
Smaller animals tend to be more numerous and to have a shorter life cycle. This helps them more rapidly evolve immunities to the diseases that afflict them and their larger relatives. Once they have evolved those immunities they can drive their larger relatives into extinction within their range, as the white-tail has done in America.
There are many examples of this. The largest trees, the redwoods, are among the last of once thriving clad with many more species. The dawn redwood lives in China, the coastal redwood lives along the coast in California, the Sequoia live up in the Sierra Mountains in California. So none of them has close relatives in their environment that can carry a disease that can kill them.
Similarly, the largest birds are part of Palaeognathae, a primitive branch off of the bird family tree. Living members of Palaeoganthae include the ostrich, cassowary, emu, kiwi, and the extinct moa birds of New Zealand and extinct elephant birds of Madagascar.
The largest spiders are tarantulas part of the second earliest branch off of the spider family tree.
The largest fish are sharks, a primitive group of fish. Other living fossils among the fish include the arapaimas, the largest exclusively freshwater fish in South America, the alligator gar, the largest exclusively freshwater fish in North America, sturgeon, and the coelacanth.
It is a long list that could be made much longer, but the point here is that the tuatara belongs on the list. The tuatara looks like a lizard and fills what would normally be a lizard niche. Furthermore, it is larger than any of the fifty to a hundred species of lizard endemic to New Zealand. New Zealand has more lizard species than any other similar sized temperate environment. So the tuatara is the last of an ancient evolutionary line that has not only survived, but has survived inspite of having lots of competion.
My argument is that it has done this because it is not as subject to disease as its competitors, and it is larger. The lizards could not evolve to sizes as large as the tuatara because the diseases carried by the smaller lizards would have wiped them out. The tuatara could not evolve smaller species because the lizards were actually superior. Without the advantage of greater size the tuatara could not compete.
As mentioned above the lizards of New Zealand are all skinks and geckos. Geckos are famous for clinging to walls, which they would not be able to do if they weighed too much. This may keep geckos small.
The world's largest skink, which is from the Solomon Islands, is smaller than a tuatara. So it could be argued that skinks are naturally smaller for some reason, even though I do not know what that reason is.
Many lizards that live elsewhere, for example, many types of monitor lizard or iguana, do grow larger than the tuatara. So there is nothing about lizards in general that makes them unable to grow larger than tuataras, at least outside of New Zealand.
It could be argued that tiny lizards are more likely to successfully raft on floating vegetation to isolated islands like New Zealand. So the New Zealand lizards might be small because they evolved from small lizards. But as they evolved so many species in New Zealand one would think they could have evolved larger species too.
We can tell several different stories and there are likely to be several factors, as there usually are several factors.