Ok, so let's look at this from an in-real-life chronological angle, and see what we come up with.
If we go with the idea that this alternate version of "Yesterday's Enterprise" would have been produced the same time as the actual episode (1990), then the last time we would have seen the TOS crew was a year before in STV: The Final Frontier. So the writers of the episode would have had to come up with a date sometime after TFF when the Enterprise-A would have disappeared. They would also have had to come up with a reason why nobody else from the TOS crew other than Kirk would have been on the ship when it disappeared, if they went with the idea that they'd only focus on Kirk. But if they used the whole TOS crew, then they would have had to explain why all these geriatrics are still serving with Kirk on the Enterprise (which is why it would have to take place not long after TFF.) And they would have had to come up with the circumstances of the ship's actual disappearance.
So we can't use anything that came after 1990 production-wise as source material, because having YE be about the Ent-A would have changed everything production-wise going forward from that point. So essentially it would look like this:
1. 1989: Star Trek: The Final Frontier: If YE only focused on Kirk, then this would presumably be the final on-screen appearances of Spock, McCoy, Sulu, Chekov, Uhura and Scotty. Since McCoy had a cameo in EaF, he would be the only character whose fate we would know. And if YE focused on the entire crew, then this would be the final TOS film.
2. 1990: "Yesterday's Enterprise" featuring Kirk on the Enterprise-A. Essentially the same plot, we have the Enterprise-A with Kirk aboard being thrown forward in time to an alternate future where the Federation is at war with the Klingons, and alternate Picard deduces that they need to send the Enterprise-A back to the late 2280's to prevent the war from ever starting, despite it being a suicide mission. If the entire TOS crew were involved, then the writers would need to figure out things for all of them to do in a 50-minute episode to justify them being there. And at the end, the Ent-A and all her crew return to the 2280's and imminent death. So no TUC, no Relics, no Generations, etc., etc. because we now know that nobody survived the A going back, except inexplicably for McCoy because he was already established as being alive at the start of TNG. And even if they only focused on Kirk, then he'd still be dead when the Ent-A returns, so the only story they could have done was a modified version of Relics where Scotty would already have known that Kirk was dead based on this version of YE. And, let's not forget, the entire plot of Yar (A TNG character, not TOS) getting a second chance of a more meaningful death is completely erased from this version, which was one of the dramatic selling points of YE.
None of these scenarios sound all that interesting to me. Kirk would have completely overshadowed everything, only to find out that he's not even interacting with the TNG crew we know, just alternate versions of them, and then we'd find out that Kirk and/or the entire TOS crew die horrible deaths when they return.
So even if this hypothetical scenario actually happened, I believe it would have turned out much worse that what we actually got. And what we actually got was pretty damn good.
