Yeah Captain to Captain like team-mate to team-mate. Think of it as PVP match and one team-mate rushes in without proper cover and gets downed and the other team-mate follows into save the first team-mate and gets wounded too which cost them the match.
Pike said to Kirk to "get out of there" but Kirk instead looped to attack. Speaking of delayed firing... during the Farragut's looping maneuver Kirk never fired a single shot until he had his ventral phasers bearing which took 16 seconds to happen. That delay cost him the Farragut as it allowed the Romulan to charge up and fire it's plasma bolt.
We know Pike was out of position as he was on the other side of the comet and he is out of firing range when coming to save Kirk when Kirk requests for Pike to fire. Pike says they're out of range but Kirk says it could draw them off. Pike orders Mitchell to fire even though they are out-of-range. Mitchell thinks their phaser hit was sheer luck. The Romulan gets another plasma shot off and damages Enterprise's weapons control which hobbled any attempt by Pike to go on the offensive. So the fiasco is on Kirk, IMHO.
KIRK: Something isn't right.
SPOCK: The captain is correct. We should be seeing them by now.
KIRK: Enterprise, I don't like this.
SPOCK: Reading gravitational signature again. It's behind the Farragut.
PIKE: Kirk! Get out of there! Red Alert!
KIRK: Red Alert!
PIKE: Erica close that distance as fast as you can.
KIRK: Attack pattern Tiberius 4.
KIRK: Get behind them. (The Farragut loop instead steers head first into the Romulan.)
KIRK: Fire! (16 seconds transpire between "Attack pattern Tiberius 4" to "Fire!")
(Farragut gets hit by plasma bolt).
KIRK: Fire phasers, Captain.
PIKE: We're out of range. We're just as likely to hit you as them.
KIRK: Maybe you'll draw their fire.
PIKE: MItchell, full phasers, now. (Pike issues orders 5 seconds after Kirk says "Fire phasers, Captain")
MITCHELL: Holy... We actually clipped them. (Enterprise fires phasers 7 seconds after Kirk says "Fire phasers, Captain")
PIKE: Nice shooting, Jenna.
If Kirk got the Farragut out of there they could of then teamed up to go after the Romulan. Instead the Romulan got them individually.