Well, I think all 3 of them ( spring, spring/side wall of a barrel, hammer/membrane and plate) make clicking sound. Animated gif at Qwerters shows buckling action of Japanese A01 ( it's not a minuature, but a cutout of actual stem/barrel of A01. and buckling mechanism of A01 is completely same with M/F ).
if you compare clicking sound of F and M, you may notice there is a big deference in sounds. A hammer of F hits circuit board on a steel back plate directly while that of M hits 3 membrane sheets on a steel plate over rubber mat( neoplain sheet?). Deference of switching components makes huge difference of sounds. Thus we can understand that a clicking sound is produced through hitting action of a hammer onto switching plate. but this is one of factors of clicking sounds.
I think there isn't noticeable difference in buckling action ( and components ) within a barrel between F and M. Sounds generated within a barrel might be basically same. As we can see A01 animation, a spring actually hits side wall of barrel, then there should be some kind of sound.
This is the 2nd factor.
Lastly. as you noted in your previous post, we can mute clicking sound using just a little dose of silicon grease around a spring. And it explains that a spring itself generates sound when it buckles.
As for the title of this thread, lowpoly's suggestion would be better for removing a coil spiring. But I'd like to recommend not to use springs you pulled out. You're removing a spring because it's something wrong. Once a spring loosened or bent, it may cause unstable buckling action even if you amend it as well as you can. Better to get a junk model M as a donor for springs, stems and caps, controllers and so and so.
In order to install a new spring.... you need a special hand made tool.
But a bamboo stick will also do this work.