The Apple store couldn't figure this out? You have system sounds muted.
System sounds can be muted and controlled two different ways. The screen lock rotation can be controlled in the same manner as well.
Settings>General>Use Side Switch to: Mute System sounds. If this option is selected, the switch on the side of the iPad above the volume rocker will mute system sounds.
If you choose Lock Screen Rotation, then the switch locks the screen. If the screen is locked, you will see a lock icon in the upper right corner next to the battery indicator gauge.
If you have the side switch set to lock screen rotation then the system sound control is in the task bar. Double tap the home button and in the task bar at the bottom, swipe all the way to the right. The speaker icon is all the way to the left. Tap on it and system sounds will return.
If you have the side switch set to mute system sounds, then the screen lock rotation can be accessed via the task bar in the same manner as described above.
This support article from Apple explains how the side switch works.
I don't know if you covered this problem with your "reset back to basics" but I had this problem after I deleted sounds/ songs from my device that were matched up the alerts.
For example I had a song that was for my ringtone, that song was part of a smart playlist.
One day that song was automatically pulled from playlist, and my phone just stopped ringing.
I should have asked my question here instead of going to the "genious bar" at the Apple store. The solution was so simple! This little button beside the volume button. It's either lock sound or rotation. I've done a complete reset for nothing!
Thanks for your concern. Hope that you will find the solution for your problem to!