There are a few issues with swapping motherboards alone. First and foremost, Face ID will no longer function - the front camera assembly in the housing and the earpiece/sensor assembly on the display are both paired to the motherboard and will disable Face ID if they don't match.
So in order to replace the motherboard and keep Face ID working, you'll need to also buy the paired earpiece/front sensor and front camera assemblies that go with it from the original phone. On the other hand, if you're okay with going without Face ID, then swapping the motherboard alone will do what you want.
In addition, the True Tone function may also be compromised. I'm not completely up to date on the iPhone 12 models, but when replacing a screen there's a file you need to copy off the original screen onto the replacement screen in order to keep True Tone working. I assume a motherboard swap would have the same issue; you'd need the data from the original screen to copy to your existing one in order keep the True Tone function. I'm fairly sure that's the way it would work, but maybe someone with more experience in that area can confirm or deny that.
On newer phones like the 12, Apple has also implemented more pairing with the logic board; specifically the screen and the battery now. While they'll still work, apparently you'll get a notification that one or the other might not be genuine - even if it is a genuine Apple part. Apple has a self-repair program that I understand can address those issues, but I'm not conversant with the details myself. However, as far as I know the actual functions will still work correctly despite the warning.
Hopefully if there are more restrictions, other members with more detailed knowledge will add to this thread.