It's sort of a meaningless and subjective question to ask how valuable a proof or construction is among others if they are all equivalent and they all lead to the same theory. I think, however, it is worth asking whether some proofs and constructions have more pedagogical value than others. (For example, the topological definition of continuity, while true in the limited domain of AB calculus, has almost no pedagogical value until the student is ready to tackle college level analysis.)
My main question is: What are the pedagogical advantages of using Dedekind cuts to construct the real numbers as opposed to the classic approach of metric space completion of the rationals through Cauchy Sequences, or Rudin's construction of the real field by assuming the least upper bound property? I have noticed that some people have strong opinions on the value of Dedekind cuts ("...the stupidest thing ever..."), so I guess what I'm asking for is a defense of the approach at least in analysis pedagogy, or even what new perspectives/theories Dedekind cuts raise that Cauchy sequences or L.U.B. do not elucidate. (This is also for my own education, as I have been taught using Cauchy sequences and L.U.B.)
I apologize if this question is not suited for this site.