New Windows 8.1 bug appeared after some update. Do another cold reboot and PIA VPN will appear again. But the bug will appear again. NVT ERP was the 1st one to disappear for me, I thought it was because I was using beta version. But then I've had Secure Folders and Emsisoft AM disappear too
I believe it might be the same for you too.
That's odd, i've used PIA in Windows 10, in an Acer Aspire (exact same conditions) for months without any issue.
Do you have any software that may be detecting it as malware or blocking it from changing TAP adapter driver? Some AV, or some software like Kerish Doctor?
Also, I remember PIA didn't show up in my start screen programs, I had to fetch it from the programs (or programs x86, can't remember) folder. Check if it's there. It may be installed but not showing up.
voodooShield used to block some PIA needed service for me. So it may be that. I cannot remember the name of the executable but it didn't sound like it would belong to PIA. A quick Google was what made me realise it was needed for PIA.
Sorry that I can't remember the name, but maybe try to disable voodooShield, reinstall it and see if it still disappears when you shutdown the computer