Ray Darr,
I can only speak from the flight deck portion of the EXIT VISA issue, i have not heard how it is for any other department ( F/a, engineering, ground staff, ) but if you are speaking of the flight deck, you are incorrect on the Visa. Actually you are only half correct and that is, you do need an exit visa to leave everytime but you can purchase a MULTIPLE VISA so that you can leave anytime you want to go.
I am not sure if this procedure has changed but as of a few months ago you could buy this permit, which took two days to be in your hand and was valid for a period of 1 year to go in and out as you pleased. If you don't buy this permit then you are correct, you need an exit visa but this procedure is for every ex-pat working in the State of Qatar. Qatar Airways have made it easier for the flight deck crew to come and go at a moments notice if they want by getting the approval to have this MULTIPLE EXIT VISA for cockpit crews.