To ring or not to ring?

I have used a variety of copper mats in conjunction with weights & washer or reflex clamp & washer on a variety of TT's and in my experience hard mats such as stainless or copper always sound better with a proper clamp system designed to couple the record to the mat. Without a clamping system the results lack focus.
Listening to a variety ( copper, stainless, original & resomat ) on the mongrel version of the Krebs Technics SP10 in the same system showed me that none of the mats were perfect. Based on my own experience I believe this was due to the lack of a clamp with the metal mats.
Same result with the Goldmund Mat, which I have heard on the Goldmund, Sota, Oracle & Technics SP10, it sounds much better used in conjunction with the reflex clamp.
Most weights manufactured today do nothing unless the record mat/platter is slightly concave because they have a flat bottom and cannot apply pressure at the perimeter of the record label to clamp over a washer placed under the record.
Very few records are truly flat, and therefore unless you are using a soft mat such as felt or leather then the records are not supported across their entire surface on a hard mat such as metal or acrylic unless a clamp/weight is used and has been designed properly, that is either the record mat is slightly concave as in the goldmund, or a washer/reflex clamp system is used.
I am not sure why anyone would put a clamping ring under the record, unless of course they are superman and wear their undies over the top of their trousers.

The primary purpose of a ring or vacuum clamp is NOT to remove warps but to more tightly couple the record to the mat or platter surface. When a stylus runs in the groove, it is actually elastically deforming the groove and the groove then rebounds. This creates a lot of vibrational energy imparted to the record. The tight clamping of the record to the large surface of a mat or platter dampens this energy and also drains that energy off for dissipation in the much larger mass of the platter. This accounts for some people finding the sound to be a bit dead or overdampened (a matter of taste and system matching).

I have a table with a vacuum clamp (Basis Debut). In use, this is far easier than a ring clamp, and, I don't have to worry about accidentally hitting the cartridge with the clamp or the needle getting snagged at the clamp/record interface. If I did not have a built in clamping device, I would use the ring. I like the less hashy sound of the play with clamping than the slightly more lively sound of play without the clamping.

The energy picked up by the stylus creates a subtle echo effect similar to hearing music in an auditorium. This distorts the original recorded music. It is also deleterious to sound retrieval. If one's system is good enough at extracting the music from the grooves, one should find it more pleasurable hearing the recorded music and the aural sounds of the environment in which it was recorded.

With regard to playing warped (even slightly warped) records: Why optimize the VTA setup only to have it altered during playback with a warped record? Is the purpose of listening to recorded music to hear what was recorded or to hear an altered version of the music?

I have used record mats (copper, alloys, Delrin, carbon fiber, in some combination) with center weights, and various periphery clamps. I have applied them to belt-drive, idler drive, direct-drive, and rim drive. When the devices are implemented, they always produce a truer image of the originally recorded music when contrasted to non-applications of the devices.

The ability of the stylus tracking the groove is analogous to walking: Is it easier to walk in a straight line on a stable surface or a vibrating one?

Larryi/Redglobe
Your posts are spot on. People should think of mats as a transmission device, not an end point. They should drain the unwanted energy away from the vinyl into the platter and beyond.
Copper is much quicker than acrylic in terms of draining energy, but equally important in determining the effectiveness of the mat is what is underneath the mat.
On my Final Audio Parthenon TT the 4.5kg copper mat,1.8kg stabiliser, 15kg platter & substantial gunmetal subplatter are specifically designed to provide a rigid platform for record playback and also wick away the unwanted energy. The whole platter system is terminated with a 40kg rigid slab of SPZ which is a superplastic zinc alloy that grain slides at a molecular level when excited by mechanical resonances.
What I hear with this deck over my earlier decks which included WT, Oracle, Pink Triangle, Roksan & Vaccuum Sota and other decks such as the direct drives SP10/L07D is an absence of smear, a focus and density to notes that is missing from most TT's. I put this down to the energy control design as well as the power supply which consists of an AC motor driven by split sine/cosine wave generators and 80w power amp. This was an early 1970's cost no object product that preceded the Melco's, Micro's and SP10/LO7D but in my listening experiences is superior.