What’s the chime type, and is it compatible? That’s what I don’t know, and it’s what I think you really want to know, because Chime Controller doesn’t connect to the transformer: It’s supposed to connect to a compatible chime.
I suspect the chime might be the issue here because of this part of your post:
This makes it seem like it’s one of the chimes that rings based on the state of the circuit. In other words, if someone closes the circuit by pressing a simple button, that supplies power to the chime and the chime will ring as long as power is applied. Once someone stops pressing the button and opens the circuit again, the chime stops. Is that accurate?
If that’s the case, then part of your solution may involve replacing that chime with a compatible mechanical “ding-dong” chime.
The other part of this system that you probably want to replace is the transformer.
How did you determine this? That’s not the rating of the transformer in the photo.
The “LOAD” label just tells you which side of the transformer is the “load” side where you can connect your low-voltage AC wiring. This is in contrast to the “line” side which connects to your standard household ~120 V AC. Your transformer’s rated output appears to be 10 V (VAC), 5 VA. (It’s difficult to tell your transformer’s apparent power (VA) rating based on the photo because that number is obscured by two wires, but an online search of the model number (MT-70a) indicates that this is a 10V 5VA transformer.) Video Doorbell v2 wants to see a minimum 16V 10VA transformer:
An existing wired mechanical doorbell and chime is required to install Wyze Video Doorbell v2. The existing circuit must have a minimum voltage of 16V-24V and power output of 10VA.
This is probably a good time for me to mention this: 
I am not an electrician. 
I have done a lot of troubleshooting with my own Video Doorbell v2, however, and I really like the product.
Thank you for including the photo! That’s very helpful! 
What additional information can you supply regarding the actual chime? A photo of that might be helpful, as well. I’m also curious about this part:
What wires did you move, and what was the wiring situation before you moved those?