capacitors burn at discharge due to low current-limiting resistor; how come they don’t burn at charging, when there is no current-limiting resistor?
The DC source has some internal resistance, aka it can't physically source infinite current (even if you directly short the terminals). Additionally Caps have ESR (Equivalent Series Resistance) that also limit current.

Do note that sometime it is prudent to put current limiting resistors at the gate of MOSFETS for example, to reduce current spikes.