Where are you getting this ball park number of 100A? He's not turning on industrial motors in a factory.
With over 30A per amp at turn on, those 5A fuses would pop even though they are slow blow.
It is the inrush current of the discharged main filter capacitors. Basic capacitors 101.
There are any number of online calculators that one can use to come up with similar numbers or one can do the calculation.
My example is based on 30,000 microfarads, because the positive and negative sides charge at different times for a bipolar power supply.
I used a transformer secondary voltage of 60 volts, although I did not see B+ or B- voltages on the schematic, because the main filter capacitors are shown to be rated for 75 volts. The surge current shown in my example will be in the ball park when compared to measured values for known power supplies.
You have to do the math for the type of fuse. See the chart below.
Since we are talking about 60 Hz power, we have to look at the time period for half a cycle and when the switch is operated in terms of the crest factor of the AC mains sine wave.
The chart above shows that the 5 amp fuse will hold over 100 amps for about a half a cycle (.01 seconds) and hold 50 amps for about 5 cycles (0.1 seconds).
To be sure these are just ball park numbers, but they are close enough for the purpose of this discussion.
Even if I drop the the transformer secondary down to 40 volts which will put about 60 volts on the filter capacitors, the surge current only drops down to about 90 amps for my example.