I have to disagree with statements as broad as some recently made.
I've seen far too many multi-pole breakers that were nothing but a grouping of ordinary single-pole breakers, with a rivet below and a handle tie above, keeping them together. GE is one brand that comes to mind.
There is no code, or UL, requirement for an internal trip mechanism. All that is required is that all legs open when a fault is induced on one leg; this might simply be a matter of the spring on one being strong enough to work three handles.
That. perhaps, is why not every multi-pole breaker boasts of having an internal common trip.
That said, at least once I've encountered a multi-pole breaker that did not have all poles open. Even with that same breaker, I was not able to duplicate this mis-action.
As for breakers where not all poles close .... well, nothing is perfect, and then it's time for a new one.
Handle ties are listed, so using them to 'make' a common disconnect is clearly allowed.
When such a tie is removed, leaving three independent single pole breakers - I'd be hard pressed to be able to prove that was ever done. I don't see a problem.