Clint, I've put kudos on your post, because whilst I discovered the same thing myself, you wrote it down before I did.
But just to confirm, yes, I also had problems with the power button not waking the screen up. Didn't matter whether it was a quick press, a long press, lots of quick presses, lots of long presses, mixture of long and short presses, power button and volume up or power button and volume down...
The only thing I could do was connect a charger or any kind. It could have been the USB cable plugged in to the mains, the USB charger plugged into a computer, or the charging dock DK26. I repaired the phone on Tuesday but had the same thing happen again on Thursday.
I think in my case the app that caused it was MapMyRun because I was out running at the time, and I normally press the power button a few times to stop it locking the screen. I have the screen timeout at 30 seconds, but the lock timeout at 30 minutes. Once that 30 minute lock time had been reached, I couldn't turn the screen on.
When I finally got back home it was well over an hour later, I was still trying anything to get the screen to turn on. When I connected it to my charging dock, the screen finally came on - but it wasn't locked, I didn't need to input my pattern to unlock the phone. Every time the screen went off, only the charger would bring it back to life, so I did a repair on the phone.
Then on Thursday it happened again, once again, well over an hour, screen not locked, but this time, as soon as I closed MapMyRun (just by pressing what I suppose is the "alt-tab" button, and sliding it away -- not doing a force stop on it), the problem disappeared, there was also a message in the notification box which said "MapMyRun is preventing STAMINA mode from entering sleep" (or something like that). So I figured out it might be something to do with MapMyRun and Stamina mode both being active at the same time. So I added MapMyRun to the whitelist and sure enough so far so good.
So whilst i haven't fixed the problem by deactivating stamina mode, I found that adding a particular app to its whitelist solved the problem for me. Of course, for other people, it's probably a different app that's causing the problem, but it's still the same solution, either add the app to stamina's whitelist, or do what you've done and completely disable stamina mode for the time being.
And yes, people with regular Xperia Z's can have this problem - that's the exact phone I'm using - an Xperia Z C6603 (the LTE version)
