A common method of controlling inrush current in some power supplies is to fit an NTC thermistor in series with the supply. At switch on, it is cold and has relatively high resistance, so limits inrush current during the initial cap charging phase. The thermistor soon warms up due to the current, and switches to it's low resistance state where it does little to impede the current flow.
With one of these, if you interrupt the power long enough to significantly discharge the reservoir caps, but not long enough for the thermistor to cool down, then on startup you get the full charging current without the current limiting. POP.
Though, to be honest, I don't recall having seen this method in anything I've had apart lately. Most seems to just have a "direct online" rectifier/capacitor arrangement which as pointed out almost certainly takes the rectifier beyond it's normal specifications. But bear in mind that most semiconductors do have a higher rating for "pulse" vs steady state - there are several limits, but one is thermal and much like a cable can carry a high current for a short period or a small current for a long period.
So as already pointed out - for these systems the rectifiers will already be "warm" when you ask them to take the full inrush current to recharge the caps. Thus you are relying on their thermal capacity when it's already partially used.
And then we get into more esoteric issues.
Most PSU chips have some sort of startup sequencing - even if it's only as simple as "once we have sufficient input then startup". Sometimes with a short power "blip" it's enough to drop the power levels to the point where "stuff" stops working, but not enough to trigger the normal power on, power up, reset everything cycle - so stuff "locks up" or behaves oddly until you power it off and on again.
Then there's capacitors, which most people don't realise have current/power ratings !
When used in power supply type applications, there is significant ripple current - which flow through the ESR (effective series resistance) of the cap. Ie each capacitor isn't ideal, it has resistance in it's leads - and most importantly - it's internal construction (long strips of aluminium foil for the types discussed here). You'll see some devices specifically marketed as "low ESR" specifically for power supplies. But regardless of that, in the detailed specs will be data regarding the allowable ripple current.
So again, in use they will be warm internally due to the long standing ripple current. If you momentarily blip the power, they get to discharge, then rapidly charge, while still "warm" and thus having less thermal capacity available to take the heat created by that inrush current.
And then some devices (incandescent light bulbs, valve heaters, CRT heaters, are ones that come to mind) take more power when cold.
But regardless of this, there is thermal cycling every time something is turned on or off. Turning off is usually more benign as the heat "just dissipates". At turn on, stuff goes from cold to normal operating temperature - usually fairly quickly. Each thermal cycle physically stresses components.
Probably none of these are enough in their own right, but when you add them all up, you get a statistically higher chance of failure during power on then when running at steady state.
