> I don’t think you can get the SCT from a log without the log publishing it?

Well you can, but only if the log cooperates.

That is, the SCT is merely a promise to add the certificate to the log by some particular deadline. Nothing technically stops the log from disregarding that promise. But if it does fail to log the certificate, the SCT itself is proof of its misbehavior.

And it's not enough to subvert a single log, because browsers require SCTs from multiple distinct operators in order to consider a certificate valid.

The SCT may be proof that a log was supposed to be issued, but browsers don't submit every SCT they see (did you know "anonymizing" proxies send CT log companies a sampling of all your TLS traffic? fun stuff!), so in practice these logs aren't audited. As far as multiple operators, almost all of them are based in the US, so just pray the USG doesn't want to target you.