My MacBook is a Battery Denier

Before you perform any hardware replacement, you might want to check the connection points on the laptop where the battery comes into electrical contact. Make sure everything is secure and clean.

There might be a part you can replace that goes between the motherboard and the battery connection point. Almost anything is cheaper than an older mac motherboard replacement. Assuming you are comfortable with this, have you taken the laptop apart and looked around to make sure anything isn't loose or melted/damaged? It sounds like it was something that was starting to fail and now is broken.

If you brought this to me for repair, these are the first things I would check from what you described in your initial post.

The battery connecter that is used on the MacBooks is very specific. Did you notice when you bought the replacement part if the pins were exactly the same configuration? If they are different that is why it may not have fixed your issue.

There is an Energy Star and a Non-Energy Star version of the battery connector.

Unlikely, but worth also checking/replacing is the DC-Inboard/Magsafe connector. This is the part that the connects the charger externally to the logic board internally.

If both of these are replaced with the proper parts, booting from a known good OS, testing with a known good battery and known good power adapter and it still does not work, then I would say it needs a new logic board. Based on the cost of that replacement and age of the machine it would not be worth repairing.