For what it's worth, LED UV's aren't as nice as fluorescents (in my opinion) especially not for Tron which shows the bare bulb.
In my case, I've always run a switching power supply in my cabinets, the kind that real cabinets use, and then either wired the PC power directly into the cabinet power, or added an accessory plug, and plugged the PC in. Everything else is powered off of the switching power supply's 12v or 5v outputs. The PC's have been the kind that turn on automatically, and the same with whatever monitors I'm using.
But, you can do as you say, just use a power strip in the cabinet and have the PC turn on automatically if possible.
For shutdown, you may want to wire a PC power button into the front of the cabinet. I used to have on on the inside of the coin door so no one else could mess with it. On my cabinets I always built them like they'd be used in a retail environment... solid... to that no inattentive drunken guest or obnoxious little kid of guest could do much damage.