Andrew,
I visited St. Andrews last February and found it very easy to get onto TOC. Granted, it was blowing 40 mph + steady the morning I had to play, but it was in the 40s and the sun was out. I think there were only 4 people on the course until about noon when we were finishing. Also, according to the locals I spoke to, Jan/Feb are the lowest time of the year as far as tourism. I can confim that, as I think I may have been the only tourist there during my two days (or so I thought).
As for the mats: They are fine, the only issue I had was keeping it secured to the ground (I used a tee) in the high wind. Come to think of it, I had trouble keeping plenty of things in one place in that wind: my ball, my golf bag (!), head covers, etc.
Funny side note: I traveled solo up to St. Andrews from London (there visiting a friend) to see the town and play the course (hopefully w/ some locals in February!). I waited on the first tee for awhile alone hoping someone would come along. Eventually a gentleman and his son arrived, but come to find out they were both from Michigan (which is where I'm from). Laughing for the first couple holes about the irony of it, we get caught by a single on #8. We ask if he would mind joining, and he agrees. His accent immediately sounds too familiar, and he introduces himself, telling us he too is from Michigan.
I traveled half way around the world to play TOC in February in 40 mph winds with a few gents from my home state...go figure