TPMS Not Learning?

Hello, it sounds like you're getting the system into it learn mode, but there is some service info here on how the system works, and also some potential reasons why the new sensors are not being learned. Such as interference with the RF signal or the tool not being close enough to the sensors on aftermarket rims. But I'll post what's here and you can go through it. Where did you buy the sensors? are they OEM parts?

This is for all the codes you have setting. If you have anything plugged into power ports (ie, phone chargers, USB devices, etc.,) take those out while attempting the relearn procedure since this is solely based on radio frequency. But see if any of this information helps you, and I will do a little further research to see if there is anything else that will hinder the sensors from the procedure.
You should have the codes cleared if possible, as well, I don't see if codes in memory will cause the system to prevent relearns, but it does for other vehicle systems.

Here is the entire Learn procedure (diagrams 1-3) and the sensor matching (4,5). Was there an issue before replacing the sensors?

Yes, the sensors were bought from eBay and they are OEM GM sensors. I recently bought them and changed all four sensors. Before, changing the new sensors, the TPMS light was on/Service Tire Pressure Monitor message when I initially purchased the car. I recently bought new tires and figured it was best I changed all the sensors with it. I will try these procedures and see if anything works and let you know.

I'll try to find the OEM part number for the sensors, eBay sells a lot of parts that appear to be OEM but are not. When it comes to any sensors for vehicles, always get the OEM from the dealership, it might cost a little more, but then you're sure to get the actual Buick/GM OEM parts. But the fact that there was a TPMS light and message to begin with may not be the sensors, do you still have the old sensors just in case? This might be an issue with the Tire pressure indicator receiver module which looks to be located under or behind the rear bumper, if that's the case and it's exposed to the elements, I guarantee there is corrosion in the connector. I don't know why they put modules where do they do sometimes, like under a vehicle, but they do, and if you live in an area where they use salt on the roads during winter, it makes it even worse. But I'll find the exact location for the module, since all for sensors are not learning, its most likely not the sensors. Is this your scan tool in the pictures you posted, the Autel?

This is the sensor replacement procedure just so you have it and want to verify they were installed correctly, which I'm sure you did. And the 6th diagram is the OEM part number for this vehicle. You're getting the first horn chirp so it's going into programming mode, at least from the BCMs viewpoint.
It appears that the Remote-Control Door Lock receiver is the input for the tire pressure sensors, and it's listed as module K77. But there are multiple T10 Keyless Entry Antennas all around the vehicle, this is all just service info on the system, but if you still can't learn the tire pressure sensors, it be a number of issues, such as the incorrect sensors, the relearn tool, or an antenna not working, although I don't think it's an antenna issue. With there being multiple antennas around the vehicle for the keyless entry, they are all looking for the same frequency to unlock the doors for example. The tire pressure sensors send out a different frequency.

Okay, I did find 2 Technical Service Bulletins on the TPMS. Diagrams 1-4 is the first one, it has to do with aftermarket devices causing interference, and Diagrams 5 to the end are for the second TSB.