For Scotty's bringing of Preston to the bridge to literally make no sense, it has to be devoid of reasonable, even if not wholly flattering, subtext for why Scotty would deliberately do such a thing, and I don't believe that it is.
On the contrary, there are certain things strongly suggested, if not implied, by Scotty's actions, that do make sense. One, he knows that Peter is a goner. He's taken a look at his radiation badge, or he saw the events that overdosed Peter with his own eyes. The scene
wouldn't make sense if Scotty were delaying medical attention, so we can rule that out. Two, Scotty's emotionally devastated, and he's passed his limit for what he can take. Three, and this is perhaps the most important, Scotty's angry at Kirk. He wants to make sure that Kirk sees first hand what underestimating the situation has cost. With the ship still in danger, that's not a spurious, vain, or self-serving end, even if it is out of line. Their lives are still all in jeopardy, and Kirk had better wise up, if they are going to get out of it. Scotty surely knows exactly what happened with the shields, and how Kirk had bungled the situation by ignoring danger. Perhaps Scotty thought he was going to read Kirk the riot act when he got in the elevator, perhaps Peter himself mumbled Kirk's name, which screwed up Scotty's courage to take the elevator straight to the bridge, but obviously any urge to say something had passed by the time the doors opened.
An alternative to going to the bridge being Scotty's idea alone is that perhaps Peter asked Scotty if he could speak to Kirk one more time before he died. Perhaps all Peter wanted to say is that he doesn't blame Kirk.
Are there other ways that these character beats could have been made, without doing that? Sure. Could any of the alternatives have worked better? Possibly. But that's not my point. My point is that the idea that it makes literally no sense, and that it exemplifies implausible behavior or behavior that's out of character, is overstating the problem; in fact, it's doing the opposite, by defining character, even if it's in ways that make Scotty look bad. Also, accusing Scotty of leaving his post is an utterly unsupportable accusation.
For an example of something that makes much less sense, refer to
Kirk and Rand holding each other as the Romulan plasma weapon closes in, in "Balance of Terror".