If a center weight is used that is very heavy, let’s say 2 lb. the lighter records will lift from the mat. This happens because the mat depression edge will act as fulcrum. This information tells us we should use a center weight tuned for the record thickness and weight. However this is impractical. Here is the solution: Use a center weight that weighs 8-12 oz . This weight will work with all but the lighter records. The alternative to a weight is the screw down clamp. These clamps have pluses and minuses. The plus is down force on the record can be controlled. The minus is if not designed properly (unfortunately most are not) spindle energy is coupled into the record. It takes very little intrusion of external energy to cloud the mechanical output of the stylus. (I wrote a paper on proper screw down clamp design about 25 years ago.)
George misses an important element of clamps, which is that to gain the most out of them quite often a spindle washer of some sort is used beneath the LP so that the clamp can dish the LP against the pad surface. This can also be used to reduce warp.
I generally don't use the clamping aspect of the clamp I use (Basis; one of the better clamps made), I just use it as a weight. Its easier.
Many years ago I saw a demonstration of how profoundly a mat can affect a turntable. A friend of mine who has been involved in several damping products over the years (Analog Survival Kit, the damping rings used by ARC, the Ultraresolution Technologies damping platform) spent some years developing a platter pad. He arrived at a fairly high degree of refinement with the mat. What local audiophiles found was that any turntable that could accept the mat (due to height and weight; it weighed about 3 pounds) sounded better and also sounded better than nearly any turntable that lacked the mat.
This seems to hold true to this day. IMO/IME most mats out there don't seem to do the job as ideally as I laid out in my opening post (IOW it was a statement of the engineering principle behind the mat, not actual execution). This is why you see so much variance in opinion about the topic.