^ It's true that Picard is the least impulsive, but that also means that he's the most deliberate out of the bunch. So, situations that don't require a lot of quick decision making are ideal for him. That doesn't mean that he can't make quick decisions, just like Sisko and Kirk being better at it doesn't mean that they couldn't and didn't deliberate over things that required deliberation.
Each one of the three has something special about them that the others don't have or don't have so much, and conversely each one of the three doesn't have something as much as the other one or two.
Picard is the natural diplomat out of the bunch. So, he likes to sit down and get comforable and talk. He wants his counterparts to be comfortable for talking too, because I think he usually expects the conversations/negotiations to take some time. This is where he shines, I think, the most because that's what diplomats do. Plus, you know you'll get to have tea with cream and sugar lumps, and maybe even some cookies too.
None of the other captains do that. Picard is ideal for diplomatic missions.
Sisko is the natural "get it done" guy. He's the one you want if you've got a lot of things happening all at once, and they are happening fast, and you need results. He had a space station to rebuild and then run, a people to be an Emissary to, wormhole aliens to deal with, The Dominion war and all that that entailed, plus a son to raise on top of it. He's more or less like, "I've got a schedule to keep, so this is what I/we can do for you, now what can you do for me/us?" Sisko is ideal for when the $%#! is hitting the fan.
And Kirk (and all of this is my opinion of course) is the smooth snake charmer. I think out of all of the captains, he's best at manipulating things/people to get a certain outcome (when possible). Kirk is ideal when you need someone that is good at "maneuvering."
I think all 3 are pretty equal at keeping their crews safe, but that's just me.
Spook/Hoorah! Admirer wrote:
^ He may be arguably approachable (though, I'd count other captains as approachable as well), but he's not the most competent. So, you might get his "help," but it might not actually be help, if you know what I mean. Still, I've got no problems with anyone picking Archer. It's a personal choice.
Oh yeah, everyone would be in terror as the self destruct sequence counted down on the Enterprise and you'd be sitting there thinking, nope not gonna die today.
Except.. maybe Kirk got beamed out at the last second by alien benefactors and you went kaboom.
Oh, I don't think Kirk is that sleazy. The captain is responsible for his/her crew, and to give the crew the slip to save himself would be... Well, he's not that sleazy. I think he would put the crew first.