There's nothing here to do with big dI/dt spikes, or what the power supply can or cannot 'handle'. Nevertheless, I think you've got the essential operation of the circuit right.
When T1 is turned off, no current flows through R4, and the voltage across C2 falls to zero because it discharges through R3. So both ends of C2 are at +5V.
Now let T1 switch on. At the instant of switch-on, C2 still has 0V across it, so the input to the Schmitt trigger doesn't change. But current now flows out of C2, through R4 and T1 to ground, and over time, the voltage at the input to N1 will fall. The time constant is C2 * (R4//R3), which is almost the same as C2 * R4 alone.
If T1 switches on and off a few times, with a period of just a few milliseconds, it doesn't really make much difference to what happens at the output of N1. C2 charges up whenever T1 is conducting, and it holds its voltage when T1 isn't. R3 is too large in value to significantly discharge C2 over that period of time.
Net result: N1 output goes high when T1 has been switched on for long enough, in total. It doesn't, however, toggle many times just because T1 does.
When T1 is finally turned off, then C2 slowly discharges through R3, and eventually N1 toggles back again. The circuit is now reset ready for the next keypress.