Could corrosion on these connectors cause shutdown during boot-up?

I've been trying to figure out what's wrong with my 2012 13" MacBook Pro. It has proceeded strangely. For a while, I had to use a USB keyboard because it didn't register keystrokes. Then I didn't have to. I probably did few zaps of NRAM and SMC resets along the way, but I don't know what fixed that issue.

I ruled out RAM but not motherboard.

One constant is the following cluster of symptoms: the indicator in the MagSafe plug is always green, but the machine will not run off battery. Disconnecting MagSafe turns it off immediately. Connecting MagSafe causes, almost immediately, the fan to start and booting to begin. It will start to boot off a boot partion on the HD but not make it all the way. Same for internet recovery. I make it through the entire status bar, see the Apple logo alone on the screen, and then the computer shuts down completely.

I have more troubleshooting to do, but before I proceed, I'd like to know something about what I discovered when I removed the battery, intending to try booting without it. There is a connector in the middle of the underside of the trackpad that shows some corrosion. After seeing that, I noticed that a nearby connector was also corroded, and some dried droplets of something on the underside of the battery, looking like it entered at the keyboard-adjacent edge of the trackpad. I was having trouble with wild pointer syndrome just before putting the machine away for a while, come to think of it. I don't recall spilling anything, but it's been a while so that means little.

Would corrosion on either of those connectors keep the computer from booting all the way? I guess I can find out by cleaning them, if I do it well and harmlessly, but I didn't think of that until just now. Also, what does the second connector I mentioned connect?

Depicted: spots on surprisingly filthy battery, connectors with corrosion.