I feel completely silly. I have repaired so many different things over the years, as well as troubleshooting settings. I’ve never imagined that my MBP to suddenly stop charging would be caused by a battery optimization setting. For anyone in this situation, please first check in System Preferences under Battery. And make sure that “Optimize battery charging” is not selected before buying parts. Haha. Plus side, problem solved!
I've had this problem with PC laptops that have built up static electricity and I've had to discharge them by removing the battery and pressing the power button for 10-15 seconds to discharge any build up. The best I can come up with for a Macbook Air is to reset the SMC (System Management Controller) since the battery is not easily removed. This is also listed on Apple's support site for possibly helping with your issue: 1. Turn off MacBook Air. 2. Connect the power adapter and plug it in. 3. On the LEFT side of the keyboard), hold down Shift, Option and Control. 4. With the other keys STILL HELD DOWN, press and hold the power button. 5. Wait 5 seconds, then release all keys. 6. Push power button to turn your MacBook Air back on. Please let me know if this helped.
I also agree with it being not necessary to remove the complete front panel. I just kept it attached and kept it at an angle as I pulled with gentle, continuous force (at a 45 degree angle as suggested) on the battery pull tab. It eventually came loose, I set the new battery in and everything works perfectly.