I think they should just be played on as the traffic is generally so low, provided that the club also scraps all competitions at the same time to reduce the potential of high volume on a given day.
We are lucky not to have any temporary tees or greens. The course is hardly ever closed unless there is heavy snow. We run a full schedule of qualifiers all through the Winter (between 7 - 10 per month)
I reckon we have some of the best greens in Hertfordhire, and they are always in excellent condition come Spring.
I put this down to some fantastic greenkeeping staff.