... The rectifier is a Russian 5C4S, which can pass 122 mA according to the data sheet. ...
The Russian data sheet says "140mA" for rectified current, but this is steady-state DC. A little below that, it says "Peak Plate Current 1.4A".
5Ц4M data sheetWhat does that mean?
During the brief window when the incoming rectified voltage is higher than the cap's voltage, there is a brief, large current pulse, as shown below:

... If there is not a bunch of series resistance between the PT and rectifier tube, those current pulses will cause the rectifier to arc plate-to-cathode. ...
There are no resistors before the rectifier.
So, it must be something to do with the specs of the PT+rectifier.
The winding resistance of the PT is also "series resistance" because it is in-series with the loop from
Cap Negative --> Ground --> PT Center-tap --> PT winding-end --> Rectifier Plate --> Rectifier Cathode --> Cap Positive
______________________________________
What governs the current?
Step back to Ohm's Law first: Volts / Resistance = Current
Now consider that the size of the Peak Current Pulse is proportional to the difference between "discharged volts" and "rectifier output volts." Or "how low does the Red Line dip before the Yellow Charging Window?"
Say it drops 50v, and the PT's winding resistance is 35Ω: 50v / 35Ω = 1.43A
Now let's say you haven't attached this power supply to any load, you have simply assembled it on a breadboard.
With no load, the cap volts are not pulled down, and there is no "Peak Current Pulse" because the cap does not need recharging. 0v / Small Resistance = Zero Current ----> 0v / 35Ω = 0A
But later when you use it to power a real amp, the too-big caps will pull too high a peak charging current (unless PT winding resistance limits the current).
Small caps (small "C") require a smaller-charge (small "Q" and smaller-current (small "I") to reach a given voltage.