Electronic – Why does a cap in front of a choke input suply reduce rectifier current? How to calculate optimal value

First, I think it's more likely these ancient tubes are no good anymore than the capacitors having failed. Except perhaps for the power, which is easy enough to check, these capacitors are probably wrapped foil or something else that is dry and should last a long time. Start by checking the supply voltages. Those appear to be well marked. If the power input diode has gone bad, then nothing else has much chance or working.

As for the circuit, I am somewhat confused too. I'll take a rough stab at it anyway. To really understand it would take more time working thru it than I want to spend on it.

The left tube seems to be a pretty straight forward amplifier. The amplified signal appears on the plate, which is then coupled into the power stage thru C4-4. Most of the mess between C4-4 and the control grid of the right tube looks to be a tone control. That's just from the general form. I haven't actually analyzed it. I think R5 is likely some sort of tone control. I'm less sure about R4, but R4 and R5 together may be something like bass and treble controls.

The strange part is how the two output transformers are hooked up. I'm guessing that the top two speakers are meant to be tweeters, the lower two the rest of the sound range, and the strange connections between the double transformer is like a crossover network. This also leads some credance to R4 being a treble control since its signal is driven from the feedback from the top transformer output.

C4-9 and R4-7 feed back a bit of the signal at TP3 onto the cathode of the power tube. This looks like classic negative feedback to provide predictable gain and a flatter frequency response.

The section of the circuit you show here can be easily enough tested in isolation. First, make sure the two power supply voltages are as marked, then feed a signal into the line you labeled as audio input. That should be clearly audible on the speakers.

For reference here and because it may change, the circuit you are currently asking about is:

This is supposed to regulate output current, but the complaint is that it is unstable. R1 is meant to be a high side current sense resistor. You say this is for charging 12 V lead acid batteries. You don't say what current, but probably a few amps. In that case 100 mΩ seems rather large. Note that at 5 A it will dissipate 2.5 W.

However, the large current sense resistor should only make measuring the current easier. It looks like your intent is that OA2 provide a ground-referenced voltage proportional to the current thru R1. That concept is good, but the implementation is flawed.

What you need is a "diff amp" that has some finite gain. The differential part eliminates the common mode voltage on R1, but the finite gain part is also important. As it is now, OA2 is being used open loop as a comparator. It's output will quickly switch between full high and full low as the current goes slightly above and below the regulation threshold.

Another problem is that the top of M3 is not connected anywhere, so it can't source any current onto R7. I don't know what that dashed line is supposed to show. Usually if things are connected to it like that it means a conductive case, but you show nothing else connected to it. A case is usually grounded, which is certainly not what you want the source of M3 (strange designator for a FET) connected to. It also makes no sense that you need to buffer the output of OA2 amplifier. I didn't look up a LM358, but if that does not have a push pull output stage, get one that does.

All in all, I'd lose the wierd current sense amp circuit as it is now. There are diff amp chips that do what you want directly. Sometimes they are called instrumentation amplifiers. These have a truly differential input, finite and sometimes adjustable gain, and the output can be referenced to some other voltage like ground.

Once you have a reasonable ground-referenced voltage proportional to the output current, you can feed it into the negative input of OA1 as shown. However, you have to make sure that the controller (OA1 in this case) is slower than everything else in the system. I mentioned this already in another one of your questions. Put a cap between the output and the negative input of OA1 to slow it down. This may require a resistor between the current sense amplifier output and the negative input of OA1 so that the cap has some impedance to work against. Do not under any circumstances attempt to slow down the current sense circuit. That will only make things worse.