I'm I wrong or the Zaelot and the Flagellant have "career abilities" that cease to function when switching to a career without an Insanity Slot Talent?

Per a strict read of the cards, yes, they'd be useless if you move to a career that doesn't have an insanity slot.

That said, seeing as how these abilities are each "once per session", it probably wouldn't break anything too horribly to let a PC use those abilities with any insanity they have, and not just insanities that are slotted in. For this too even come up, the PC has no doubt been suffering from one or more insanities for an entire career, so it's not like they haven't paid their dues, so to speak.

Assuming the GM was cool with it, that is. Some GMs might hold out instead, hoping that a later expansion will provide additional careers with insanity slots, and I can't say that I'd take issue with that decision.