2013 MacBook Pro 13" A1425 - POWER ISSUE

Model: MacBook Pro (Retina, 13-inch, Early 2013) - A1425
MacOS: Catalina 10.15.7
Processor: 2.6GHz Dual-Core i5
Memory/Storage: 8GB/256GB
Battery Info: (See attached pictures) Cycle Count: 1, Condition: Normal

Problem TLDR: MacBook is not working unless charger is plugged in - battery appears to be in perfect condition.

Backstory: The original battery was starting to balloon, so the previous owner (who is tech-savvy) bought a new battery from Amazon. The replacement battery was tested (before using the adhesive) and the laptop would turn off the moment it was unplugged. It also wouldn't turn on unless plugged in. The owner then bought ANOTHER third party battery, and this issue persisted. The owner also informed me that he tried the original battery again and still same result.

What I've done so far: When I bought this laptop to fix, the battery icon in the top right said "79% Battery Not Charging" (MagSafe 2 Light Amber). Previous owner said it wasn't going up no matter how long he left it plugged in.
- When I took it home, first things I tried were SMC and PRAM reset - no change.
- Then I decided to take out the battery and inspect the battery/logic boards - no visible water damage.
- The DC-in board was a little wonky with the MagSafe 2 charger so I decided to unscrew it, and re-seat it. I noticed that it might have had a bit of something sticky on it at some point. I cleaned the one side with IPA - couldn't check the other side as this would require I take out the logic board and I haven't done that yet. The pins on both the 3rd party 60W charger, and the DC-in board look perfect.
- I put the battery back in, turned it on - there was a change! Now, as per attached picture, the Battery is at 100% and says "Battery Is Charged" (MagSafe 2 Light now GREEN). But the laptop still turns off when I unplug it.

What's Next?: I know the easy answer is to get another battery, but I think there is enough evidence here to first explore other alternatives. The odds of two third party batteries in a row being duds isn't 0%, but it seems unlikely. I'd like to exhaust my options, opting for possibly buying the $20 DC-in Board before $130 on an iFixit battery. My thoughts are that the battery is in good health based on the laptop recognizing it, and the fact that it's new, but the MacBook is not properly switching between Power Adapter and Battery. Is this a known possibility? Is this something that the DC-in board controls?

I plan on removing the logic board for complete inspection in the next day or two and I can update on anything I find (or don't find). Please let me know any information or relevant experience you have with this situation. Any information on how the DC-in board works, what controls the switch from charger to battery, what parts on the logic board to inspect... ANYTHING. I'm just getting into laptop repair in the past month and I'm really excited to learn.

EDIT 1: In response to Dan requesting Coconut Battery Photos
Had to get an older version of Coconut Battery(3.6.6) because I'm only running Catalina, hopefully no differences in info though.

EDIT 2: Testing done 2024-12-10 with the MagSafe 2 Charger plugged in.

1. The first thing I tested was the Fuse F7040 which protects PPBUS_G3H. I got around 12.3-12.6V on each side of this fuse which confirms PPBUS_G3H is present and the fuse is good.
2. R7050 coming from battery: 12.3-12.6V on each side. This I believe confirms the battery is providing voltage correct?
3. From here I didn't know what to do next but I tested MOSFET Q7800... this looked like it had 6 pins (3 on left 3 on right) but the .brd has the component in a different orientation with 7 pins. Either way I got 3.3V on what I believe to be the Gate and Drain sides, and then 0V on both of the Sources? The 0V was on top right and bottom left pin if your orientation is with the battery closest to you. If true this would mean that the MOFSET isn't switching properly?
4. I wanted to test if the SMC control signals. I took the board out and I went to U5010 which is labelled SMC PWRGD. This component has 9 pins, I tested 2 so far. Pin 1 which is PP3V42_G3H has 3.3V and Pin 5 which is SMC_RESET_R_L also gives 3.3V.

This is where I ended as I was just getting more confused about what I'm supposed to be looking for and was hoping I could get a little more direction. I will also note that on Step 4 after I took the board out (no SSD or battery) and plugged it into again, the fans would run absolutely wild for a bit before stopping. The heat sink wasn't at all warm but the fans would go crazy - is this normal?