1. The rebuilding of Yeshivos, Baitai Chinuch, and Bais Yaacov’s is no less an expression of Avavas Yisroel for every Yid then sending Shluchim. It was (and is) and disagreement regarding what you is the best meathod of Harbatzas Torah.
2. Demanding Halachic proofs against Mashichist beliefs is besides the point. If you make an error regarding a Halacha in Shulchan Aruch or Ramabam that’s one thing. But let’s say you believe in something for which there is no direct positive evidence and insist that it is your religious duty to believe so based on personal feelings, revelations, miricles, etc. You have just legitimized the revelation narrative of every world religion and cult.
3. I take personal offense at the statement made by CS that the Vilna Gaon saw the Alter Rebbe of Lubavich through a key hole, realized that if he would meet him he would have become a chossid and ran away. But this is just a symptom of the above stated problem.