Click-and-pop suppression when circuit is not yet powered
Red is system +5 V supply rail, and yellow is my device's 3.3 V buck converter powered from the system's 5 V. You can see the power surge when switch is being turned on, and 3.3 V power supply also false-starts.

Sanity check:

+5V and +3V3 show very similar damped ~500kHz oscillation on scope display. In fact, almost exactly the same signal considering one is 1V/div and the other 2V/div. It's very much like both probes are measuring the same thing.
However, +3V3 is the output of a Buck converter. Presumably there are enough decoupling capacitors on +3V3 rail, since when the buck is on and delivers +3V3, there is no visible ripple on the scope:

So, the buck's L and C do remove the switching noise, yet the ~500 kHz damped oscillation at turn-on which is present on +5V would somehow pass through the same buck LC filter and end up in the output unattenuated?... This is not realistic.
Besides, you got 2V/µs slew rate on the damped oscillation ; on a properly decoupled rail this means huge transient current which would have to come from somewhere.
So. You're measuring pretty much the same signal on both channels probing unrelated power rails. So they must be connected pretty solidly at &quot;highish&quot; frequency. The only HF connection between these rails is not through the inductor, but through ground and decoupling caps.
So, I think you're not measuring what you think you're measuring. There's probably something there, but it is not a 2V/µs, 500kHz damped oscillation on both power rails at the same time.
It could be a interference from the switch arcing or something else, being picked up by the scope's ground alligator clip lead acting as a loop antenna. So you can use the tiny ground spring instead.
Or it could be common mode noise that the probe and scope interpret and display as actual signal.
The way to check your measurement is to repeat it, for example repeat the measurement in your first screenshot, make sure you get the spike. Since the scope triggers on Channel 1, now move Ch2 probe from where it was to somewhere nearby on the ground plane, and repeat the measurement. Ch1 should show the exact same signal as before, and the scope should trigger. Ch2 should show a 0V flat line since you're probing GND. If it does not, and it shows a spike that shouldn't be there, then your problem is either a common mode voltage spike at turn-on, or something which generates a large current pulse in the ground.
Or perhaps you've connected your ground alligator clip in the wrong place...
If you are still using the ground alligator clips, you can also unclip Ch2 ground alligator from the circuit and clip it on the tip of Ch2 probe tip, then repeat the measurement. This will measure what the loop antenna is picking on. If the scope displays a spike on Ch2 with such a setup, then you should really use the tiny ground springs and not the alligator clips.
If this is true, then the problem is not what you thought and the solution you're working on (a muting circuit) won't solve it.