That link is broken for me, but since it says doorbell A/C transformer, I presume it's ok. If you have a multimeter, you might just check at the input terminals of the doorbell to make sure you are seeing 24 VAC there (make sure you put the meter in A/C mode). Outside of that, it should be charging. Now as mentioned there are issues with charging efficiency when it gets to cold temps. Depending on settings, it's possible that it is not able to keep above water on charging if the number of events per day is high, even at somewhat higher temperatures. In my case, with a couple of zones set up, and seeing ~20 events per day, plus a couple of doorbell presses, I was seeing the battery loose ground over time starting at around 35F (1.6C) or so. Below that point it would steadily drop down to the point where it would shut off. With some adjustment to reduce events, I've been able to get it to stay alive steady state down to around high 20's F or so, but it looses ground below that point.
The net is, it appears to only trickle charge the battery using the incoming A/C. It doesn't run directly off of A/C. See some of my posts in the above mentioned thread for more about why that would be the case for this device. There are other devices that can run directly off of the wired doorbell A/C, but they are mostly lower resolution devices, don't have the on-board AI, etc. It's an unfortunate situation. It's not clear yet whether some firmware tweaks might be able to help improve things, or not. I have my doubts they'll be able to do much on this., but I'd be happy to be proven wrong!
