Across Z1 13.9V
Across Z2 9.9V
-Seems reasonable. The are the supplies to the control circuit. I would have thought these were 15v and 10.5v zeners so maybe a little low but probably OK.
Across Z3 4.3V
A little low, should be 5v. TTL min volts is 4.75. Probably won't matter though.
Across Z5 5V
Z5 would appear to be a reference. It's fed from a constant current source, VT3 and Z6.
Across Z 6 0V
Should be 3.3 methinks. Might be a problem here. Although, Z5 has a sensible voltage so are you sure about Z6?
Across pins 7 and 14 of IC2 4.2V
Provided by Z3.
Across Pins 4 and 8 of IC1 24V
About as expected. It's the sum of the + and - rails on Z1 and Z2.
The NE5558 is an op amp. (Not to be confused with an NE555) Op-amps can typically handle up to 30v, so not a problem.
As for the waveform on IC2/8, be aware that this whole circuit is on a floating ground, connected to pin 13 (I think, number is damaged) which is the series pass output. It's possible that no point in the DC circuit is grounded, so if your probe ground is on the case you are just picking up induced hum in a floating circuit.
To check this waveform you need to use two probes set up to identical volts ranges and both on the same zero line, one on IC2 pin 7 and the other on pin 8. Then set the scope to subtract mode.
Some care is needed with placement of scope grounds on such high current equipment. If you put the probe ground on the wrong place you could burn out the scope.
Anyway, I would be more concerned with whether VT4 and VT5 are being turned hard on by an earlier fault, or if one of them is s/c. Disconnecting R21 or pulling IC2 would eliminate any earlier fault. If R27 is still very hot, you have your problem area. A voltmeter is all you need here to measure Vbe and Vce.
Never overlook the possibility of more than one fault. However, you need to tackle things in stages. First, find out why R27 is overheating.
As to how it works, IC1/1 is an error amp which compares the voltage across* the series pass unit with a reference, Z5. IC1/2 is a comparator which changes state when the voltage on C5 exceeds the output of IC1/1. C5 is charged via R13, providing a time dependent voltage. When the rectified AC goes through 0v, VT1 turns off, allowing VT2 to turn on and discharge C5, restarting the 'timer' consisting of C5 and IC1/2
Thus, the voltage on C5 increases progressively through the half-cycle. When it exceeds the output of IC1/1 the comparator fires, activating IC2, which provides SCR firing pulses. By varying the comparator reference voltage, the error amp can exercise 'phase control' over the AC input to the transformer to keep the voltage across the series pass unit to a few volts, regardless of PSU output voltage.
* This is why the circuit has a floating ground, because it measures a voltage (across the series pass unit) which is not referenced to ground.
BTW I would check that pin 12 is not connected to some wrong place. Since it's directly on a TTL gate with no series resistor, if that got a substantial voltage on it, it would knock the hell out of that circuit. Might explain Z3 burning up.