Codes P2101 and P1516, engine goes into reduced power?

That's with the connector unplugged I assume? And this is with the new TB or used one?

You can back probe the connector wires with T pins or back probing pins if you have them, I'm looking up the information on a battery disconnect for this vehicle, disconnecting batteries is not like the old days, and this vehicle might have a smaller aux battery for Onstar options or Stop/Start system. Plus, there can be procedures that need to be done after a battery disconnect such as idle relearn, along with having a battery current sensor on the negative battery terminal. Some of those have a nut on them that can be a one-time use. So, clearing the codes with a scan tool is really the correct way to do it. What scan tool do you have? Is it just a code reader? I just don't want to see you disconnect the battery and end up with other issues.

No it's not a good idea to move the throttle plates by hand, these new electronic throttle bodies should be opened with a scan tool. When cleaning the throttle bore and plates I always open the throttle plate with either a scan tool or have someone sit in the vehicle with the key on and press the throttle down to open it, I know some people will open them by hand, but in some cases that can damage the internal components. The new ones are very complex, you may have two bad TBs, if the used one doesn't function with the key On, and came from a used parts lot, who knows how long it's been there or if it got water inside. This is really where a good scan tool comes in, that way you can monitor what the ECM is seeing for voltage signals on the TPS and throttle actuator.

Sounds like they are both broken, the used one sounds like it seized up.

Once you get the codes cleared, you can follow this throttle relearn procedure, this is the non-scan tool procedure to be done if the throttle body is cleaned or replaced. Can you read live engine data with the key "on" with that scan tool? The communications from the data link connector looks to go through the body control module then to the ECM. But at least you weren't reading any voltages on the TB wiring, you may try checking the yellow wire and the brown/white wire each (with the key Off) for any shorts to ground, but it sounds like the original TB was actually physically broken inside, with stuff bouncing around inside.

That's great news, really happy to hear things worked out well for you. Keep us in mind for future issues.