In the case of Yeti 2022 you were only able to log (pre-)certificates signed by particular CAs trusted in the Web PKI, which were due to expire in the year 2022.
In practice the vast majority of such logging is done by issuing CAs, as part of their normal operations. But it is possible (and is done purposefully, at least sometimes) to obtain certificates which have not been logged. These certificates, of course, won't work in Chrome or Safari because there's no proof they were logged. But you can log them yourself, and get SCTs and show those Just In Time™.
This is only an interesting technical option if you have both the need to do it for some reason and the capability to send SCTs that aren't baked inside your certificates. The vast majority of punters just have the CA do all this for them and the certificate they get has SCTs baked right inside it so there's no technical changes for them at all, they needn't know CT exists.
Because the CA's critical business processes depend on writing to logs, they need a formal service level agreement assuring them that some logs they use will accept their writes in a timely fashion and meet whatever criteria, but you as an individual don't need this, you don't care if the log you wanted to use says it's unavailable for 4 hours due to maintenance.
Most likely not. But the tech behind FBA (Federated Byzantine Agreement) distributed ledgers would make an extremely reliable system that can handle malfunction of hardware and large outages of nodes. And since this is a write-only log and only some entities can write to it, it could be implemented with permission so that the system doesn't have to deal with attacks that public blockchain would face.