Howdy!
To get right to the point: I had this exact problem on my 7th generation thin ipod classic. I replaced the click wheel. I am happy with the new click wheel but it did NOT fix the same play button issue that I was having. I even cleaned up the logic board and it did not help. So I think there may be a problem with the logic board. I could replace that.....
I did replace my headphone jack which was not working and BOY does it sound good now. I just use headphones that can electronically play/pause the ipod for a short-term fix.
NOTE: I don't know if it is really a logic board issue, but neither click wheel works so I couldn't imagine it is anything else!
Me