Electrical – What are these transformers in the garage for and how to disconnect them

The other transformer is for some other system - another doorbell, an alarm system or (small possibility) HVAC. So even if you can move the wires and power the NEST with that transformer, it is possible that (a) the combined loads of the two systems may be too much for the transformer, as it is rated at exactly the value recommended for the NEST by itself, and (b) there could be some unexpected interaction between the two systems - i.e., one switching on/off might somehow induce a problem into the other one.

Transformers are cheap. A quick search finds them at around $10 on Amazon, and you may be able to get one in a regular store (which has overhead but doesn't have to build in the Free Shipping to every little product) for less. Just make sure it is UL Listed as one side of the transformer is connecting to 120V AC.

It's OK (desirable, actually) if the new transformer's VA is higher. The transformer VA rating indicates how much load the transformer can safely support. The doorbell VA rating indicates how large the doorbell load may be. It's important for the voltages to match and for the VA rating of the transformer to be higher than the VA rating of the loads (the sum of the chime and the Ring device). This is analagous to payload capacity of a vehicle: the actual load should not exceed the carrying capacity.

Since tracing the low-voltage doorbell wire back to the transformer isn't working, maybe you can work the other direction. Turn off all circuits in the building except the one that powers the doorbell. If the circuit isn't marked, turn off circuits one at a time until the doorbell doesn't work anymore.

With the supply circuit identified, and with only that circuit powered, you could trace that circuit with a the tone pickup tool you've been using, a non-contact voltage tester, the live wire warning in a stud finder tool, etc. "In theory" by inspecting every connection along that circuit's path you'd eventually find a pair of light-gauge stranded wires spliced into the circuit somewhere. Those would be the leads of the existing bell transformer.

Even if the transformer is mounted somewhere awful, like on the back side of a steel outlet or light junction box behind a finished wall or ceiling, at least its mains power junction should be accessible and therefore disconnectable.