No, because they don't have to: if they stick to the proper tee boxes, they'll equalize themselves. Plus the odds of a course being five yards short on EVERY hole is basically nil. Again, so long as they stick to the proper tee boxes. Keeping records would be busywork.
It can be equalized by being 90 yards long the next day, but do courses really keep track of such settings? I doubt it very much. It's very frustrating when the white tees (example only) are consistently set up short, often in a foolhardy attempt to improve pace of play (people are responsible for slow play, course set up won't improve it to any great extent).
It's OT, but I disagree that yardage can't help slow play. It too has to be more extreme than five yards to make a difference, though, so maybe that's what you're saying: a few yards won't matter.
Those who play this shortened course consistently then end up with lower indexes than their capability and complain when players from other courses consistently whup them in net events. For handicap indexes to truly reflect the player's capability, the courses must be set up to average the rated length over a relatively short period of time (maybe every week).
Shoot 84 and your index is off by about 0.3 if you play a course with about a 139 slope that's 90 yards short of its rated yardage. It's still not a very big difference.