It seems the key here isn't—or shouldn't be—what kind of service the defendant used, but whether something special happens when a service is involved in preparing a message to his lawyer.

IMO if the "for my lawyer" purpose/intent is not in dispute, then it shouldn't matter whether the service is a search-engine, an LLM, a browser-based word processor, or the drafts/sent folders of a webmail client.

The reverse direction is much clearer: Imagine a client receives an obviously-privileged email from their lawyer, and uses a cloud text-to-speech service to listen to it. Should that audio/text be admissible as evidence? Hell no.