Chrome Sync is, AFAIK, not a push service. Something polled a Google server, it returned a bad answer, it crashed the browser. Why is this important?

Because it's entirely possible that Firefox or Safari, for example, could have been crashed by contacting the safebrowsing server, and the safebrowsing server returning an answer that crashes it.

Firefox also does remote firefox update checks and plugin update checks, etc.

None of the browsers you mention are "independent" of internet servers anymore. They are meant to function independently, as is Chrome, but exactly the right remote bug could likely crash all of them.