Stop right there. The stories are not consistent with each other.
Should have? Maybe. But DIDN'T.
We don't know that. What we do know is if that story is consistent with Past tense then for a while, the federation DID disappear as the result of Kirk's future being uncertain.
The federation disappeared because Gabriel Bell died in 2024 caused by interaction between the transporter beam and the cloak on the defiant. It was fixed later (and is rather paradoxy, as sisko and Bashir couldn't go back in time had the federation never existed, meaning bell wouldn't be killed, meaning the federation would exist, etc)
If the orb of time never existed it still would have happened.
Kirk never died in Trials, in any timeline. There was originally the one where "I don't know sir" was a redshirt, then there was one where it was O'Brien, but both universes were similar enough to be the same - Kirk neve entered formal reprimands on anyone's record for that fight.
kirk is very important to history, in many ways, but not because of the orb of time. Kirks actions in Trials were identical to his actions in a non-orb-of-time universe, and while he could have been killed in trials, he could have been killed in any number of times.
Sisko and crew didn't choose to use the orb of time to go back and endanger Kirk, they were pulled back against their will and saved the day.