I will give you my two cents. First, it is easier if your furnace has a common(C) terminal on the controller. My 22 year old system does.
Mechanical thermostats, what we had prior to modern electronics, were simple temperature sensing switches, those systems could function with two leads, one powered conductor up to the thermostat, one back. Connect the two and the furnace comes on.
Old people may remember the mercury blob thermostat, it was a coil of bi-metal foil that expanded, rotating as the temperature changed, the rotation moved liquid mercury in a glass vial to bridge two pins, connecting the two leads. In my 1960’s youth my dad, a frugal engineer who grew up in the Depression, would turn ours down at night before he went to bed, then back up in the morning.
I installed a “programmable” on my system maybe 20 years ago, it was powered by two AA batteries that lasted five years between changes. I bought a Wyze so I could control the furnace from my iPad, retiring that programmable.
24 VAC seems to be the standard voltage in heating systems, that’s a legacy issue. I don’t know why they used 24 VAC. It may have been it was easy to make from 110V, also under 50V is less likely to do damage, easier to insulate, less likely to start fires. Old doorbells used a similar approach.
Why not 5 volts, or DC? It may be way back then, before solid state devices were inexpensive, that converting to DC was cost prohibitive. And voltage drop, a problem on longer runs of wire, is less of an issue the higher the starting voltage is, 24 volts may have been the sweet spot. Now we use WiFi instead of running low voltage wires, but you still need power. A friend, working at Amazon, told me years ago that Bezos implored him not to use batteries on the secret project he was running, it was Echo/Alexa.
A thinking, WiFi connected thermostat needs a return or earth to power the electronics. Wyze gets around this on old systems with the small hockey puck thing. I have not yet taken mine apart to see what is in it, but it must somehow supply a ground/earth. I am a Mech E, not an EE, but have worked with electronics a lot, I have not seen this done before.
If your furnace has a C terminal run a wire from it to the Wyze thermostat and you are done. C is common, or earth. You could get an earth somewhere else, but do that with caution, only using a VOM, and only if you know what you are doing. All earths are not always the same.
Voltage should come up on the red wire, hook it to Rc, (not Rh) white is the return, that is all you need to switch the furnace on and off, provided you can power the electronics, to do that you need a common. Use a VOM to be safe and sure. If you don’t know what that is, own and know how to use one, this may not be the project to learn on.
I personally would not use another 24V source, that’s beyond my pay grade. Before I did that I’d try to get past the AC/DC converter on the board and just use a DC supply to run the control electronics, or ask an EE who owes you a favor what the issues are.
The Wyze use of the green wire (fan) is, in my heating system, superfluous. My furnace controller delays the fan until the heat exchanger is up to temperature. I hooked up the green wire and now my fan comes on early, I plan to disconnect it.
The Wyze UI is very beta; it is poorly written, does things I hate and Alexa can’t see it. Hopefully they will get savaged in forums and improve it. Wyze lightbulbs are not much better, so who knows.
Electricity starts fires, so don’t mess with stuff you don’t understand.
Did I say electricity starts fires? It does. Really. All the time. Yes, heating systems have safety fail-safes built in, but they get very hot, use combustible gases, fire and other things that kill people and burn down buildings. Be careful. To learn play with 5V DC electronics, like Arduinos, not heating systems.