IMO: This might be a contrarian opinion, but I don't think so. Its much the same problem as asking, for example, if every single line you write, or every function, becomes a commit. The answer to this granularity is, much like anything, you have to think of the audience: Who is served by persisting these sessions? I would suspect that there is little reason why future engineers, or future LLMs, would need access to them; they likely contain a significant amount of noise, incorrect implementations, and red herrings. The product of the session is what matters.
I do think there's more value in ensuring that the initial spec, or the "first prompt" (which IME is usually much bigger and tries to get 80% of the way there) is stored. And, maybe part of the product is an LLM summary of that spec, the changes we made to the spec within the session, and a summary of what is built. But... that could be the commit message? Or just in a markdown file. Or in Notion or whatever.
I do think there's more value in ensuring that the initial spec, or the "first prompt" (which IME is usually much bigger and tries to get 80% of the way there) is stored. And, maybe part of the product is an LLM summary of that spec, the changes we made to the spec within the session, and a summary of what is built. But... that could be the commit message? Or just in a markdown file. Or in Notion or whatever.