2013 MacBook Air hot; shuts down within 1 min of power cord removed

I’m the original owner of a mid-2013 Macbook Air that’s been working great for the last few years. Over the last month, I’ve run into a new problem that’s been extremely frustrating. I’d like to avoid getting a new laptop, but I’m stuck. I poked around the forums and got Coconut Battery and TG Pro to try to help troubleshoot.

Problem: when the Macbook is not plugged in, it will shut off within 2 minutes. Sometimes it’s immediate, mostly it’s a few seconds to a minute. Battery health is good according to Coconut Battery. If I try to restart it without the power cord plugged in, I can hear the fan start up, then immediately shut down. I don’t hear the chime nor see anything on the screen. If I plug the laptop in and press the power button, the fan will start up and immediately shut down 3 times, but after the third time the computer will start up and I can use it as normal.

I use my computer pretty lightly - mainly for browsing the internet. That being said, I have seen TG Pro show CPU temperatures around 100 C (!) - especially if my CPU is high (generally appears to be when Google Drive is syncing).

A few months ago, I swapped the original 128 GB SSD hard drive for a 1 TB OWC Aura N SSD. Everything else is either original or was replaced by Apple under warranty 6+ years ago.

Would love some insight and advice. I’d really like to continue using this laptop, but the lack of portability won’t work for me.

Plugged in:

Unplugged, a few seconds before turning off: