How suppress bounce-related turn-on spikes in DC bench power supply?
You are confused about the noise getting through because your active filtering elements should not let it.

But there's no power yet, so how will any active element filter the noise?

When you have a ringing supply rail that noise can be propagated and even amplified quite easily throughout the electronics and properly predicting or modelling that is hard enough that most professionals also just prevent the supply ringing.

To do that, I would prefer having a Zero-Crossing switch on the primary side to begin with (whichever fits your circumstances best). That because it also prevents ringing noise being emitted at switch-on to the rest of the world.

Then, I might prefer also adding a low Rds-on pass MOSFET that turns on a few ms after the main power rail's capacitors are and stay above a certain minimum voltage. Maybe add a 1/10th the capacitance capacitor behind the MOSFET so that in its Turn On curve the electronics still see a "slow turn on", if you feel that might be beneficial.

Do also test turning on another original one, while your new one is in regulation and see if that ever (give it a dozen tries) causes noise as well, maybe your feedback filter will prevent that, or maybe you'll need some more filtering if you want to use it in a Lab with sensitive loads.

A cheap Lab supply at a customer had huge ringing spikes on the output when someone else turned on the soldering iron. Very annoying with $5000 a piece prototypes.