iPod A1136 Click Will Center Button Fails to Click

@lkvee here is my answer to a similar issue from 5 years ago

the center button is actually nothing but a plastic piece that sits in the center of the click wheel and when pushed on depresses a microswitch on the logic board. Sometimes the center button plastic does no longer make good contact with the switch. You can try to either replace the clickwheel/center button and see if that works. The other thing you can try is to add some depth to it by applying a few layers of tape to the underside of the button. If that still won’t work, you can measure the switch itself first. Attach the leads from your multimeter to the bottom two contacts, you should see that the switch does have some resistance, and total continuity when depressed. If the switch itself checks out okay, you may have to replace the logic board. If the switches do not work, you 'could' try to replace those. It will be difficult to find the right microswitches. You could purchase a broken iPod or a non-working logic board at places like Ebay. then unsolder the switches and exchange them. BTW the switches are the same on a 6th gen. Classic. Hope this helps, good luck.

iPod 5th gen/w/ video center button doesn't click - iPod 5th Generation (Video) - iFixit