That's not how network services work. If a person is authorized to make the API call, the person is authorized to make the API call.
Service operators are not permitted in law to dictate client software, outside of copyright license terms.
Service operators are not permitted in law to dictate client software, outside of copyright license terms.
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.
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.
If I run a service, I can add pretty much any requirements (barring anti-discrimination, accessibility laws, depending on your jurisdiction) I want for use of that service.
If I saw you can't use my service unless you do so while wearing a chicken hat and humming God Save the Queen, I have every right to do so and every right to kick you off if I discover you were wearing a turkey hat instead, because it's MY service. I may have a hard time enforcing such requirements, but nobody has a LEGALLY ENFORCEABLE RIGHT to use my service without my permission, no matter how dumb my requirements are to be granted that permission.
Note: The above obviously doesn't automatically apply to existing legally binding contracts such as with a paying customer, though in most cases either party is free to break the contact at any time assuming they are willing to pay the penalty specified in said contract.
That's not how the law works today. Weev was still tried for accessing publicly accessible URLs.