I don't think there is any case law related to "which client was used" applications of the CFAA.
I can't really imagine any reasonable circumstance where someone would be authorized with one client and unauthorized with another. It's not really a reasonable position; if you offer a free network service on the internet, and anyone is authorized to use it, I don't know of any legal basis for gating that authorization on "user of your software".
Perhaps some internet-clueless judge will one day disagree, but that's not how the mores and norms of the internet work today. Signal doesn't get to declare me a criminal if I write my own client software to talk to the Signal API servers (or use some other open source software I found), regardless of Moxie's squeaking. Either your service is public and open to everyone, or it isn't.
The Supreme Court's upcoming decision in Oracle v. Google may also be relevant from what I gather.
Just because you write it in your ToS doesn't mean that users agreed to it.
If your api is publicly readable (without any auth which you could only have by agreeing to the ToS), then whatever they write into their terms is irrelevant.
This doesn't apply here however, because you can only sync with Google servers if you have an account, which means you must've agreed to the ToS.
It's also the user that's breaking the ToS, not the developer of the competing browser... But that's irrelevant here in guess :)
That's entirely unrelated to unauthenticated and public APIs and their possible ToS relevance.
Some cases have been found to not be a violation given a public API, but in the case of sync like this, it is not a public API. It's a public facing api but it requires authentication (the keys in question) in order to be used, this means that using google's keys without authorization from google to use them for that purpose would likely be seen as a CFAA violation by the courts.
You can't have an account without agreeing to the ToS, so it's definitely a violation.
The only law I can think of that could get involved is the DMCA. If my client implements usage/content restrictions (very stupid design) you using the API with another program in violation of the terms could be considered circumventing the content restrictions and that's illegal due to the DMCA.