2013 MacBook Air doesn't turn on in certain circumstances

My friend has a 2013 MacBook Air that was working fine one day, and the next day, it wouldn’t turn on at all. We tried plugging it into the charger, and the light was orange, but when we pressed the power button, the light turned green and then back to orange, with no other signs of life.

Today, we opened it up and disconnected the battery, then plugged in the charger. The MacBook power cycled a few times, then turned on and worked normally. After that, we reconnected the battery and tried turning it on without the charger, and it worked fine.

However, here’s the strange part: after shutting it down, it wouldn’t turn on again, even after plugging the charger in. We unplugged the battery, and it powered on automatically, but even after shutting it down then, it still wouldn’t turn on unless we unplugged and plugged the charger back in (which caused it to turn on automatically).

In the past, the MacBook had some odd power issues, like the battery showing 95%, but then randomly shutting off. Now it seems like the problem "evolved" into something worse.

We noticed a small red dot on the motherboard, which in online photos of the Mac's inside is usually white.

Any ideas on what might be going on?