Trivial vulnerabilities are easily discoverable yes -- but, they are also trivially discoverable by standard automation available today. I've found GPT-4 to be shockingly bad at vulnerability analysis for all except the most popular vulnerability classes. My speculation is that there just isn't enough literature on these vulnerability classes for it to have practical mastery of them.
Complex vulnerabilities are the emergent phenomena of multiple events across a codebase and it's dependencies, involving control flow, data flow, while missing type information and other runtime data. Even Anthropic's 100K context windows won't nearly fit it all, and if you stuff all the code into embeddings, the ability to reason across all this space will be poor.
You can train a model to ask very pointed questions about particular snippets, but wholesale LLM-based analysis to find vulnerabilities seems like it'll be extremely slow, expensive and inaccurate.
I do not know of LLMs that have been specifically trained on, say, the testing corpus of some "lint" programs and against known vulns. As you point out, it wouldn't be possible as a user of an LLM AI to do the equivalent by showing it some vulns, while it would be perfectly reasonable to get an LLM to write, for example, business case studies by showing it examples.
Inference is so slow, and almost everything about fuzzers are meant to be super fast. Maybe there's a late stage part in crash validation/analysis where you can use it but my bias is that we're just not there yet.