So it seems like the answer to this question is pretty obvious and revolves around three primary points:

1) Wool is cheap and hard-wearing

2) Wool is a 'working-man's' fabric

3) The fabric doesn't really matter.

As to the first, it is obvious and has been stated above. Wool is cheap, readily available, and hard-wearing. Whilst I believe that Rand provided money to the residents of the Farm (as it was originally called) when he first began to establish a school for male channellers, I don't think this extended to an open check, as it were. This means that basic clothing and food would have been provided for those who wish to train to become Asha'man, but nothing nearly as lavish as we see later on in the series (looking at you Taim).

From the Wiki - "Black coats are pieces of clothing that are the mark of anyone training to be Asha'man. Obviously, the coat is black, but the material varies. Some wear wool, some silk, and some (like Mazrim Taim) have embroidered the otherwise unrelieved black."

As to the second, remember that Rand wasn't interested in creating a fancy, pompous and elitist institution like the White Tower. In fact, the Black Tower was supposed to be the opposite of the White Tower in many respects, and this was one of the most important. The original purpose of Aes Sedai had been subverted over the long years, and rather than being "servants of all" they were now aloof from the world, pulling strings and manipulating monarchs. Ash'man are above all a soldier; all training is based upon this goal, and wool is a fitting fabric for a soldiers uniform. It is as stated, a 'working-man's' fabric; plain, appropriate and functional.

From the Wiki - "Asha'man training is much more militaristic and battle-oriented than the training of Aes Sedai. Asha'man are taught to do everything with the One Power, including mundane physical labor."

As to the third, well, it really doesn't matter what the fabric is at the end of the day. The consideration of how hot the Asha'man will be is irrelevant. The trick of concentrating to ignore temperature is also taught to Asha'man, meaning that they will be perfectly fine wearing wool in high temperatures, and indeed will be inclined to learn the trick faster as a result. The point "pp. 348-349 of TPoD: "....refusing to let heat or cold touch you did not make them go away. You could ignore them right up to the moment sunstroke cooked your brain or frostbite rotted your hands and feet." is a good one. However, there seem to be a few inconsistencies in the books regarding this. We also have references to Asha'man being brought water whilst they are training, in order to stay hydrated etc. So the trick of concentration, coupled with proper care, and also the pressure to perform in demanding conditions seems to me to mean that wool is a perfectly acceptable fabric. 

From the Wiki - "Because of the concentration technique, the material chosen is based on personal preference and not on climate."

That's just my 2 cents though. 

I have no idea why this thread seems to have gotten off-topic (although maybe I am missing something), but the questions seems like it has two components.

When you say "Ter'angreal like the A'dam be used in a Stedding" what do you really mean? 

1) If you mean that a damane can still channel the one power at the direction of the sul'dam, then no, as the damane will not be able to access saidar. 

2) If you mean can the sul'dam still control the damane and make the damane feel things, then possibly yes. 

The WoT fandom site states: 

The a'dam creates a link between the two female channeller; the wearer of the collar is completely controlled by the wearer of the bracelet. Latter is not even required to be able to channel on her own, only to have the ability to learn how to channel is enough for controlling the wearer of the collar.

So the a'dam creates a link, and the wearer of the bracelet can control the wearer of the collar, so long as they can both channel (spark vs learned ability is irrelevant here). 

The problem is that is it not expressly stated that the a'dam requires any input from saidar to work in regards to controlling the damane. It is never stated that when the sul'dam wants to punish the damane, the sul'dam forces the damane to channel weaves at herself. This supports the argument that the a'dam would work in a Stedding because the abilities of the a'dam regarding feeling what the damane feels and making her feel things, is not dependent of actively channelling. Having the ability to channel, and being able to reach the source, are two very different things.

Mat's necklace would still work in a stedding, the rings of the Bloodknives could be activated and used in a Stedding, and  I would assume that a binding rod could be used, if the channeller used a well to channel the power into it. Even Egwene's twisted stone ring would work, as it only needs to be touching the skin of the user to function.

In short, there needs to be a differentiation of ter'angreal which are made with the power but do not require active input from it to function, and those which actively require the power to function, such as the Bowl of the Winds. 

Anyway, this is my first and hopefully not my last post. Let me know what you think.