I'd say the greatest what the heck moment in TOS has always been the opening of The Naked Time. When your job requires you to examine an abandoned ship in a *hazmat* suit, that's really not the time to take off your glove and start testing the random space powder with your finger. I really hope that guy was drummed out of Starfleet.
In terms of moments involving the main characters, I would definitely mention 'Wink of an Eye'. First the technical issue of contact. The only thing different between the accelerated people and the normal people is how fast they move. Which means that if Kirk really wants his people to know he's still alive, he could've easily accomplished that by just staying in the Captain's chair until they noticed him. And his last ditch effort of recording a tape with the computer could have easily been accomplished right away, on the bridge, and it would've been much easier for them to find, too. In fact, if he can interact with the computer in the first place, then he could've immediately left Spock a message by other means that couldn't even be confused or missed at all.
Far worse than any of that, though, is the ending. An entire people are trapped in this terrible condition, McCoy instantly finds a miracle cure and the Enterprise just... warps away. I thought this was Star Trek I was watching? Where's the compassionate attempt to provide them with the cure as well? It would've made so much more sense, not even just because of the Trek factor, but because it would've lead to a much better ending all around. Whether you try to bring them the cure and then realize that their time ran out while you were synthesizing it, or whether you successfully bring them the cure and leave Deela to live with the fact that she killed people on the strength of this idea that their condition was 'incurable' only for 1 Federation doctor to succesfully cure it in a matter of hours - this ending could've been incredibly memorable and poignant instead of 'huh?'.
Actually how much are these guys accelerated?
Assuming the maximum. Kirk was accelerated for say 50 hours and Kirk was missing for 1 hour. So 1:50 at the most.
At the end Kirk was assuming that Deela was already dead wasn't he?
However each week in Kirk's time would be only a year in Deela's time. The Federation might have time to develop a "cure" that would work on Deela's people if they were interested
Maybe they wouldn't be interested. Deela's people may have killed hundreds of people by accelerating them and what did they do with the unneeded women? And the men who were rebellious like Kirk and not docile like Compton? Also if they were generation after generation more the species they'd kidnapped rather than their original race aren't they mostly "human?
More evidence that Spock is the 2nd in command of the Enterprise. He's acting like someone who has the responsibility of command when his captain is indisposed. He's got to stop Kirk attacking his crew and keeping the command structure intact while the transporters are being repaired..Well, "I could have come up with better rescue ideas than Kirk!" is pretty much the point of the entire episode. Whatever Kirk does here, it's always too little, too late...
What is really weird is the way Spock concentrates exclusively on protecting Kirk's reputation, instead of tackling the problem down below. Then again, he's but the Second Officer here; apparently First Officer Scott is doing his best pursuing the solution closest to his heart, fixing the transporters.
Timo Saloniemi
He is taking and delegating the work to the chief engineer. He's still keeping his hand in being a sounding board for Scott.