Doesn’t bother me, but your ‘old school’ memories are clouded. American style parts click, too. (p360 excepted, see below)
Somewhat. The clicking comes from the microswitches. There are a few sticks that don’t use microswitches though, like the Suzo Inductive, Perfect 360, and Sanwa Flash. Suzo Inductive I don’t know much about so can’t comment. Perfect360’s seem to be a hit-or-miss ordeal, with most recently manufactured ones having difficulties detecting corners properly, but they’re easily available to buy direct from Happ Controls. Sanwa Flash is the only option for in Japanese style, but they went out of production something like 4-5 years ago, and the flash assembly by itself can easily run >$200 in the second hand market. There’s rumor of some hacker in Tech Talk trying to make an optical replacement for Sanwa sticks, but nothing has come of it yet.
Can you get/make a silent stick? Sure, but expect to go American parts or pay out the wazoo. Or you can get used to the clicky. For less noisy but not quite silent options, you could try to locate quiet versions of the standard microswitches in most sticks; all of the microswitch datasheets I’ve seen include a noise volume, so you can look for the quietest replacement and switch the clicky switches out for quieter versions, but this is almost guaranteed to require a bit of wiring work.