Seems like most people include a switch in the circuit from the PID/Timer to the alarm. This way you can turn off the light/buzzing (or prevent it from triggering in the first place, say baby/wife/gorilla is sleeping). I have never totally understood the momentary timer switch, but perhaps that's how the Omega timer works. My ebay sestos timer can just be 'reset' from the interface, and w/ the alarm switch I can just kill the buzz if its bothering me.
I have wondered about using the timer to (dis)engage a pump after a set time, but if I wanted this kind of automation I'd probably be using a BrewTroller or BCS. For now, an alarm at set points which cues me to make changes (turn on a pump, set the timer, add hops, etc) seems like miles ahead of where I'm at.
If you really wanted the timer to start at a temp you could wire it up to an alarm on the PID... you'd need a switch on the timer 'power in' to change between "PID Control" and "Timer Autonomy". With the switch in 'timer autonomy' it is powered normally, timer is on and you can program a time. Switch it to 'PID Control', the PID temp alarm will trigger the timer to turn on (and presumably resume/restart the last setting) - otherwise the timer will be off.
An idea, but not something I would find helpful in my brewing. Your call.