This is a partial (might be mistakes as im on the move atm):
In a movie that generally either has stardates that the general audience cannot reconcile with and the lack of clocks present because it is the future (Star Trek V bridge for example has no clocks, and VI is a redress of that set, the only new set built being the galley), this movie has clocks added as a method of reminding the audience that there is a countdown at hand, on two counts:
First, to play along with the Sherlock Holmes mystery solving of when the torpedoes were fired, and then when the assassination was carried out and then of when the perpetrators escaped.
Second to give the audience a countdown reminder to the peace conference. If the protagonists are not there in time, another event will unfold that they are trying to prevent.
I suppose a third is the short time in which to track Kirk and rescue him from the penal colony.
That the OP thinks there are many clocks on the bridge is not by accident.
There are actually only two clocks but their placement is important. The one on the bridge is nearly always in the background of hero crew shots and important plot points, including the first indicator of an energy spike, to classic original crew line up shots, its nearly always in a reverse shot; the other clock on the viewscreen is like a news channel, it time stamps important events, and is also the view from the bad guys.
Throughout the movie there are references to time, and most of it is how time is short:
SPOCK: We will be able to follow the Captain's movements.
VALERIS: How did you achieve this, sir?
SPOCK: Time is precious, Lieutenant. We must endeavour to piece together what happened here tonight.
NANCLUS: Mister President, they are vulnerable. There will never be a better time.
SPOCK: Even logic must give way to physics. Gravity had not been restored by the time they escaped.
SPOCK: Ah! Mister Scott, I understand you're having difficulty with the warp drive. How much time do you require for repair?
SPOCK: However, if I know the Captain, by this time he is deep into planning his escape.
McCOY: Well, if you've got any bright ideas, now is the time.
KIRK: Time is the problem.
SULU (on viewscreen): I'm getting underway now. We're now in Alpha Quadrant. The chances of our reaching the conference in time are slim.
KIRK: ...then all we have is a neutron radiation surge, and by the time we're close enough to record it we're ashes.
CHANG (on intercom): You do prefer it this way, don't you? As it was meant to be. ...No peace in our time.
FEDERATION PRESIDENT (OC): The total evacuation of Kronos has been calculated within the fifty Earth year time span.
SPOCK: The key, please, Doctor. Time is short.
FEDERATION PRESIDENT (OC): ...As you know time is of the essence.
Sound track-wise, I think it is has an undercurrent of time ticking away, for much of the period from when Kronos 1 is hit, onwards.