I made a couple of simplified block diagrams to show why running the wyze thermostat requires a C wire and why if you use an external transformer, you cannot use the Cooling aspects of the wyze.
Not sure if this will help anyone, but it’s worth trying.
Current needs to flow through the R and C terminals of the transformer, through a load in order for that load to get powered properly. Without a complete loop back to the transformer, you run the risk of the current finding its way back through your high voltage wiring in your house, or through a ground loop, which can cause equipment damage and relay chatter.
Here’s a diagram of a thermostat that is being powered by an external transformer.
Note that the only load that has a complete loop is the heat, fed by the W wire.
By contrast, here is a thermostat that is being powered with the furnace transformer, with a C wire (or C wire adapter).
Once again, not sure if this will help anyone, but it’s often easier to see a diagram and understand it than to have it explained to you without a visual aid.