That still doesn't work for me. Kirk's reaction on the bridge, and in engineering were both immediate reactions to unforseen circumstances. Scotty's reaction to bringing Preston's body to the bridge is considered and time-consuming - and by going from Engineering to the Bridge, he had to pass Sick Bay. Or he was at least closer to it in transit than he was by the time he reached the bridge.
I'm not sure how you're drawing that conclusion. We never actually see the moment when Peter is injured. There are multiple shots of him trying to help his shipmates after the initial phaser volley strikes the engineering section as several of the cadets are shouting that they "can't breathe" or "need air." Scotty's voice can be heard over the commotion urging them to use the respirators built into their suits and to stay at their posts, which he later tells Kirk that only Peter does.
Without knowing when Peter was hurt, it's difficult to know how much time Scotty had to react or to consider what he was doing. We hear him talking to Kirk via the comm system, but there's no way to know if he'd learned what happened to Peter by then.
This incident reminds me of something that happened a few years ago. I'm a pediatrician by trade. One of my patients during my residency was a young girl who ended up in the ICU due to uncontrolled seizures precipitated by a prolonged illness and poor nutrition. The first of her seizures happened at home. Another happened after her mother put her in the car to take her to the hospital. It was during the second seizure episode that the mother decided to pull into a fire station to seek help, as she'd been driving by when the seizure started.
The girl was eventually transported to our hospital via helicopter after the firemen called 911 themselves. A number of my colleagues wondered why the mother hadn't called 911 herself while at home or merely continued to the hospital, as we found the entire sequence of events somewhat strange. What we concluded based on discussions with the mother was that she was so frightened by what was happening that her immediate thought was to seek help for her daughter, which she tried to do by going to the fire station.
All rescue workers- firemen included- are trained in Basic Life Support (CPR), but most do not have the medical knowledge needed to treat seizures unless they've also completed training as an EMT (I know one or two people who've done this). This is why we were puzzled by the mother's decision to go to the fire station instead of calling 911 from either home or from her cell phone while in the car. But in hindsight, I can see why she took the actions she did. She knew her daughter needed help and went the first place she could find.
Scotty's situation strikes me as being somewhat similar. He knew Peter needed help but may have been too traumatized to go to sickbay or call for help while still in engineering. Whether the turbolifts being out contributed to the problem isn't clear. We also don't know exactly where McCoy was. He was on the bridge immediately before the attack began (he's clearly seen standing behind Kirk as the latter orders Yellow Alert) but seems to be entering the turbolift as soon the the phaser strike begins. When or how he made it to sickbay isn't known. In any case, Scotty may have been under the impression that he was still on the bridge or that he could go to Kirk for help (as he might under different circumstances).
I realize it doesn't make perfect sense, and that's why people have a problem with it. But what I keep coming back to is this: not everything in real life makes sense. Movies aren't real life, but they're meant to depict real life as closely as possible (even in the twenty third century). That entails leaving some loose ends lying around and not providing explanations for every single event that occurs. Doing so makes things seem forced and contrived, which makes for bad storytelling. Now, I don't condone leaving huge plot-holes lying around, but the occasional oddity is acceptable for the reasons I've mentioned above. There simply isn't a way to make sense of everything that happens in a situation. It'd be nice if all the puzzle pieces fit. Sometimes they don't.