My understanding of how the development of cancer works is that all the error-checking mechanisms in a cell have to become damaged, before a cell can be damaged in such a way that it mutates into a cancer cell, because if they aren't all damaged before that, then one of them will find the "error" going on, and self-destruct the cell.

Cancer itself, then, is a constructive proof that DNA damage can disable DNA-transcription-time error-checking mechanisms.

The question in my mind, then, is whether those safety mechanisms are "blocking" / "synchronous" steps done during DNA transcription, such that disabling them would make DNA transcription occur faster. If they are, that would neatly explain the observed effect.

Not really, a lot of cancer mitigation is done by the immune system.