Kirk, Piccard, Aubrey and Adama
The beginning of the answer, I think, lies in two names: James T. Kirk, as performed by William Shatner. Take either one of them out, and the whole thing falls flat.
I mean Patrick Stewart, for example, is a far better actor, and Jean-Luc Piccard was a far more credible captain. Longtime readers of this blog might remember a devastating little parody that nathan bissonette wrote last year concerning the original series' chronic tendency to put the most essential officers on the ship square in the middle of the greatest danger. Piccard, on the other hand, behaved as if he understood that extremely dangerous situations are why you have junior officers and enlisted personnel.
But Kirk is the guy with the mana, the mojo. Kirk is the one that all the others end up getting compared to. Kirk is the paradigm, the prototype, the original, the one and only -- or is he?
Which brings us to the second half of the answer: Kirk, as played by Shatner. When you take a closer look at Kirk, you find that he is the bastard great-great-grandson of Captain Horatio Hornblower, the hero of a hugely successful series of novels that C.S. Forester wrote in the 1930s and '40s. (And you might also take a moment here to compare the dramatic delivery styles of Gregory Peck and William Shatner, as Peck was the actor who played Hornblower in the movies.)
And after you're done doing that, take a moment to think about Russell Crowe as Captain Jack Aubrey...