Your impression is correct (about the cold compresses)
  http://www.niams.nih.gov/hi/topics/childsports/child_sports.htm
Though this says it's for children it works for adults also.

Question one..." Wouldn't less compression improve circulation?"
With regard to those braces, not really. They aren't tight enough to
effect it circulation.

Q 2..."Or is the compression from the brace far outweighed by the
warmth?"
The compression is used more to limit mobility then to relieve
swelling, while the warmth causes blood vessels to diliate(expand). The
more blood that flows to the area the faster it heals.

So the braces are used to hold the affected area "semi" steady, and
improve the blood flow, by using heat.

I hope that helped.