The common ancestor suggestion (for that is all that it is) is based on another of Ancestry's algorithms called Thrulines. That does not use any DNA results as such. It trawls all searchable Ancestry family trees for any that contain you or your matches, or even people with similar names that it thinks may be the same person, and tries to assemble a link between you. The linked relatives and common ancestors may have been obtained from each or either of the trees in which you and your matches appear, or they may have been cobbled together from other trees in which Thrulines believes it has matched some of the intervening ancestors and has taken additional linking individuals from those trees too.
In other words, it may or may not be correct. It depends on the accuracy of the trees that it has used to assemble the information, and whether any individuals it has used to assemble the suggested link might be the same individuals it has found in another tree or trees, etc. Thrulines suggestions can be useful, but treat them with caution and do your own research on all the individuals in the proposed links between you to try and establish whether the proposal is actually correct.
Your following description of frustrations and lack of responses is not unusual. I have certainly been there, as I would imagine has almost everyone who is or has at some time or other been attempting to research DNA matches. It can often be a case of you having to research your matches family tree, from scratch if necessary, with whatever information is currently available to you, keeping an open mind and being willing to explore all avenues. Long and tedious work unfortunately.
One of my highest matches on Ancestry has ignored all my attempts at contact. His relationship was puzzling to me, as it was obviously fairly close, but I already had a fairly extensive, wide and deep tree, at least for several generations back to the 1841 census, and some lines beyond that. But the name was a completely unrecognisable to me. He has a public tree, but it only contains nine people, and straight away I hit problems with two entirely unrelated individuals who were both a good fit for his father, making further research back along that line problematic. I persevered over the course of about 18 months, during which I went down several wrong avenues and had to retrace my steps, and spent a fair amount on GRO certificates to confirm and progress my research.
Eventually, I was able to establish that one of his grandmothers was an illegitimate child, having been born quite some time after the death of her mother's husband, and that her mother was the sister of a woman with whom one of my GG Uncles had two illegitimate children, and later married. Whilst I can't be certain that he was also the GGF of my DNA match, the fictitious father named on his GM's birth certificate had the same occupation as my GG Uncle, and they lived within a stone's throw of one another at the time. There is no other scenario that fits within the time frame, unless his GGF was another of my GG Uncles who lived in the same area. Whilst that is a possibility, they seem to have been in less likely situations to have had the opportunity or motive, although I can't completely rule them out.
So I guess what I'm saying is that some matches can be identified without any assistance from them, if you are willing and able to put in the work - depending how badly you want to identify them. What may be important and/or critical to your research for one match may be less so for another, and I'm not in the business of simply collecting every available match I can find. I choose those that can either corroborate or disprove my existing research, particularly in areas where other definitive evidence may be lacking.
I had another recently on Ancestry. No response to messages, and a rather unusual username that bore absolutely no resemblance to a normal individual's name. From the 3 person public tree, I was able to establish that the match was probably female. A search of the internet for the username revealed a pinterest account using the same name, and from the public information I found there, I was able to narrow it down sufficient via GRO searches, that I was able to identify her, and within a couple of generations back I recognised an ancestor who was already in my tree.
If you haven't already, I would strongly recommend researching and assembling as many collateral lines as possible - so not just your direct ancestors, but also their siblings at each generation, their husbands or wives, and their children, bringing each line forward to as near the present as you can. It's a lot of work, but so often it enables you to recognise a link fairly quickly when you start researching back through the ancestors of a match. It helps enormously, even when your match may have a useable public tree that you can look at. So many trees on Ancestry contain information that is unsourced and/or not cited, and in many case obviously just copied en masse from other public trees without any checking of the facts asserted.
I can only wish you luck. If you want to learn more about DNA and how you can use it in your research, the pitfalls etc., I would recommend obtaining a copy of Tracing Your Ancestors Using DNA, edited by Graham S. Horton. It covers the basics through to some fairly advanced information that you may not need to use or be aware of, but I have found it invaluable.
