Almost all the circuit boards that I've repaired that were very old needed new polarized electrolytic capacitors. That is what fixed this one too. Mine 'hummed' just like yours. The flash bulb was fine in mine.

 

If you are able to measure the voltage across the capacitor that charges up that will tell you if the high voltage capacitor is OK if you get at least 150 volts. That flash tube is quite different from those I am familiar with, which are mostly "U" shaped.

 

Finally yesterday I had some more time to check the timing lamp again.
DC voltage on the capacitor parallel to the xenon tube is around 430V. It seems to be fine.
There are no polarized capacitors on the board, all diodes were checked

I got another U shape xenon tube from another timing lamp (also not working) which I bought as new 20years ago and used it maybe twice or three times. Lamp has been connected to the board and still no flashes.

How the arc in the lamp is ignited? The middle contact contact around the tube is connected to a “coil or transformer” with three contacts. I measured their 0V even the sensor has been attached to spark plug cable.
The coil or transformer is driven by semiconductor labelled with T10601.As next step I am going to check this.

 

The flash tubes that I am familiar with are triggered by a high voltage pulse to a bare wire that is wrapped around the flash tube. That pulse is created by the pulse transformer, with a current pulse fed to the primary of the transformer. On a few timing lights the pulse is from a direct connection to the high voltage to a spark plug.
On most powered timing lights the trigger pulse is sensed by a current sensing pickup clamped to the plug wire. If that pickup is not fully closed around the wire it does not work.

 

T10601 is a Thyristor,( C106D) there's usually a high voltage across the end tubes around 350V to 400V and a small transformer to strike the tube , .

 

Yes, the T10601 is an SCR that is triggered by the pulse from the clamp on magnetic pickup on the high voltage cable to #1 plug. So the direction of the pickup matters, if it is backward the SCR may not trigger, resulting in no flashes. That is the case with SUN timing lights, at least the generation that I am familiar with. I have seen others with an arrow on the clamp on pickup.
Of course, rough handling may break the ferrite core of the pickup, or damage the fine wire coil. So that can be checked with a resistance meter,

 

I just repaired an AUTOTUNE model #4115 that the light stopped flashing. It was C5 on your schematics (600V/2uF). It is located in the handlegrip. My large non-polorized capacitor (part# 1-6704-1) was shorted internally at about .4 ohms. I replaced it with a 400V/4.7uF, and it is working again. The SCR was perhaps a GE137, and the NPN transistor was RCA 1-6732 (TIP31/41 or TCG196).