In the second photo the black object you are pointing to is probably an SCR. But I do not see it in the first photo.
The second photo is taken from about the 10:00 position in the first pic. That is a bridge rectifier he's pointing at in the second. The pen in the first pic is just propping the PCB board against his Ideal tester.
In the first photo you are pointing at a 15,000 resistor with a burn spot. A 15,000 ohm resistor would logically relate to the test resistor. However, the orange strip looks more red on my monitor.
I believe that is a 1.5K resistor, and part of the low-voltage DC supply circuit, along with the bridge and the cap you identified.
Just above this resistor is a white object that may be a capacitor. Above the white device is where the black SCR of photo 2 should be. But whatever is in that spot does not look black or have the top shape of a TO-92 package.
No, the bridge he's pointing to in the second pic is behind (at that angle) the resistor and cap. There is a TO-92 transistor (or may be a low-power SCR) with a painted flat face directly above the cap. It's between us and the bridge in the second pic.
If the SCR failed open or the trip coil, then operation of the test function would keep power applied to the 15,000 ohm test resistor and thus cause it to burn out. 120 V applied to a 15,000 ohm resistor is close to 1 W. The test resistor is probably a 1/4 or 1/2 W rating.
The large resistor is a 1- or 2-watt unit. For reference, the resistor next to the 8-pin DIP IC is a 1/2-w unit. The "test button no worky" because the circuit has no power supply. It would not respond to a real accidental shock in use, either.
Where is the trip coil and mechanism and how does this circuit connect to the trip coil?
Is the trip coil burned? And is it now an open circuit?
Most likely failure of the SCR would be to short, and this could very well blow a hole in the package.
The coil and output side of the contacts (receptacle and feed-thru terminals) are on the other half of the receptacle, not shown in the pix.
Possibly, but that shouldn't damage the power supply, which is not in line with the power circuitry.
My guess is a high-voltage spike blew out the supply, or something in the circuitry has shorted downstream of it.