Here, Thanks to Michael from Zurich, Switzerland, and for the benefit of everyone with a 3562A showing similar signs of disrepair:

The 3562 A shows a fault code, is on the A2 (SYSTEM CPU/HPIB) error code 19, which means Monitor RAM Test Multiple Monitor RAM failures.
All voltages checked, and they are OK, the ripple is OK and the clocks are OK too. Everything was OK, no smoke, but I still had to solve two issues.

Issue (1)
Non-working Display Unit HP 1345A

First of all I guessed that this issue was coming from non-working A2 (CPU) but installing the test jumper to get the test screen did not work. Measured the voltages on top of the HP 1345A. These did not show any issues (no shortened tantalum capacitor) on +5V/+15V/-15V. Checked a few more voltages but not the +105V.
So I removed the HP 1345A unit to do a visual inspection and noticed the defect on the A3 board (Low Voltage Power Supply). Q1/Q2 did not look so good.

For sure without +105V we do not get any picture from the HP 1345A. I removed the faulty Q1/Q2 and solder some test cables for +5V/+15V/-15V/+105V to A3 to power the assembly from an external power supply. Lucky me, no other defect and the test picture came up.

I decide to order SG3524 (pulse-width-modulator), MJE180, and all capacitors of the A3 to replace them all. After the rework on A3 I carefully powered up the +15V supply which is used for the DC/DC converter to generate the +105V and measured the current.

See the re-worked A3 assy:

No issue seen anymore, the +105V is working. I added a 2.7k resistor to create a nominal load (approx. 37mA) on the +105V path to adjust the +105V. Just to know what was causing the burned Q1/Q2 I swapped the new SG3524 with the original one and I see that the current was increasing like hell when slowly increase the +15V voltage. So I guess the major problem was a defect SG3524 here.

Issue (2)
Faulty A2 board with hex error code 19

After installing the now working HP 1345A back into the HP 3562A I got a bit more detailed information about the A2 problem.

It says that there is a problem on the low byte SRAM on A2. To ensure that nothing else causing this problem (bus issues) I removed the A2 from the cabinet and it can be operated completely standalone. From the LED on A2 I still got the same error code as before (when A2 was installed) so at least there is no other
board causing this issue and I really can focus on the A2 board. First I checked all signals on the two 32kx8 SRAMS (U212/U211) with a scope but I did not see anything defect, everything looked so far good (no shorts, activity on all signals, etc).
So I attached my nice Philips logic analyzer.

Playing a bit with the logic analyzer, but did not get any more results so I believed what the monitor test logs said and replaced the low byte SRAM (U211) with a new one (ordered 70ns ones from Mouser).

After replace the SRAM the self test is passing on the A2 and it’s now time to install the A2 board back again in the
HP 3562A cabinet (and crossing fingers!!!!!).

With the changed SRAM, my HP3562A boots up without any other errors and issues and is ready to be used again!