I'm treating this like they're running for Federation President in the 24th Century (and somehow Kirk has been revived since his death in 2371). I'm going to use ranked-choice voting, AKA instant run-off voting, where you rank your preferences.
I'm going to write off Archer and nuKirk right off the bat. NuKirk is too young and immature; he's loads better in
Star Trek Beyond, but in ST09 and
Star Trek Into Darkness, he's basically a frat bro in a Starfleet uniform: Not ready for power. Archer, meanwhile, is probably dead by the 2380s... also he probably
was Federation President already, and may be ineligible to run again if they have term limits.
That leaves a revived Kirk, Picard, Sisko, and Janeway.
It's a fairly tough one, to be honest. Sisko is probably my favorite character, but I would be worried that his morally compromised decisions would come back to haunt the Federation if he were elected. Last thing you want is the Romulan Star Empire demanding you extradite the Federation President himself to stand trial for tricking them into going to war with the Dominion.
Kirk is obviously a great man, but he's also more of an executor than a manager. He's the guy you send to save the President, not to
be the President.
That narrows it down to Picard and Janeway. Both of them are capable, heroic leaders. Both of them are deft negotiators and diplomats, and both have demonstrated the will and strength to use force when necessary. Both of them have saved the Federation on numerous occasions. Ultimately, I would probably give the edge to Picard -- ultimately because I cannot quite support voting for a person who ordered the death of an innocent being in Tuvix. The only reason I'm not outright invalidating her from getting any rank on my vote is that she
did save untold billions of lives by infecting the Borg Collective with that virus-thing in "Endgame."
Final ballot:
1. Picard
2. Janeway
3. Kirk
4. Sisko
Archer: no rank
nuKirk: no rank
This makes me wonder: how are Federation Council reps selected? We never see any of that. I assume there are elections?
Nothing canonical. In the novels, Federation Member States each get one Federation Councillor, and they chose their Councillor in whatever manner they want. So the Federation Councillor for United Earth is popularly elected (
ENT: Rise of the Federation - Uncertain Logic), but the Federation Councillor for the Andorian Empire is determined by the governing party or coalition in the Parliament Andoria (
Andor: Paradigm), and the Federation Councillor for the Third Republic of Bajor is appointed by the First Minister with the advice and consent of the Chamber of Ministers (
Bajor: Fragments and Omens).
IMHO.
And who has executive authority? Starfleet has to be answerable to someone--doesn't it?
Official and ceremonial it's the Federation President, but (from numerous examples) operationally it's pretty obvious that the Council is what directs Starfleet's activities.
Is the Council actually giving orders, though, or is it passing laws or setting policies that the President then orders Starfleet to obey?
We see the Federation President referred to as commander-in-chief in DS9's "Homefront/Paradise Lost." He also has clear executive authority over Starfleet operations in that episode, issuing orders about deployments and security procedures on Starfleet bases and ships. He even declares a state of emergency that empowers him to place Starfleet officers on every street corner on Earth. And in
Star Trek VI, we see the Federation President being briefed by plans to retrieve Kirk and McCoy from the Klingons, but not the full Council. The Federation President also seems to be setting most of foreign policy in that film.
ETA:
In the novels, it is pretty explicit that the Federation President has executive authority over Starfleet. However, the Federation President works closely with, and presides over full sessions of, the Federation Council. And one particular committee of the Federation Council, the Federation Security Council, shares some executive authority over Starfleet with the President.