It does not make a lot of sense for me. It would have made if I would have read it in the 18th century. Today, not so much.
I know that we like to think those guys were getting BBS all the time and were knowing what they were doing, as they were using a straight, but we need to be realistic too. We have better blades, better sharpening stones, better soaps, etc. We are getting better shaves than those guys were getting back in the day. As someone was saying in that article, the shaving was painful, and I definitely believe that. Not only painful, as they were getting irritation, but painful as they were not having sharp enough instruments. With that in mind, cold water can act as a mild anesthetic. So, it might have worked better. It would have masked the pain of dragging a blunt instrument over your face. Menthol works that way too.
Rest of the things are people that try to justify something that they are doing for one reason or another. Is pretty much the same thing as with people that are separating soaps into tallow and non-tallow, even if, technically, there shouldn't be any differences.
I've tried cold water shaving, and I feel that all soaps are performing as best as they can with warm/hot water, while only some of them are keeping those qualities with cold water. Some will not lather as good and might not provide great glide if cold water is used. While, I can shave with both, warm and cold, I generally prefer warm. If I have only cold, I'm definitely not gonna skip the shave, as it's working good enough.
Of course, it is. Ice acts as an astringent that way. You can skip your aftershave and use ice. If you apply it on how/warm face, you might even see better results with ice than with something else.