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When the battery is removed does the board it’s attached to stay with the computer. The repair shop wants to keep the board. The battery replacement kit doesn’t mention a new board. Update (03/05/2021) Yes, I was able to see the pulled battery at the repair shop (I asked them to just pull the battery). It looks like Apple has designed Macs to only draw power from the battery system so just plugging in an external power source will not allow you did run the system without the battery pack. The last time I replaced a battery in a Mac it was a self contained pack and wasn’t required to power the computer.
The battery daughter board is directly connected to the battery and generally is not removable. That means that if the battery is removed then so is the battery daughter board. Replacement batteries will come with the board as it is usually considered part of the battery. You can see the board is connected in the image for this product.