I cant imagine any Rosh Yeshiva believes that his yeshiva doesn’t have something to offer more than any other yeshiva. And I have no doubt that any Bachur in a yeshiva(assuming he likes it there) believes that his yeshiva is the best. (kind of like summer camp)
That being said there are two questions to be answered. 1)why are some chofetz chaim rabbis so open about it? and 2)Why was(is?) Chofetz Chaim considered by some as a “cult”?
The answer is that in years past there were very few “non-Chofetz Chaim-ers” that joined the yeshiva. The derech wasn’t even a consideration. Why? Because it made no sense to one who is not familiar with the benefits of the derech.
If someone wanted to go to college and then work, why would he go to a yeshiva that constantly talks about the greatness of Harbatzos HaTorah?
If someone wanted to go into Harbatzos HaTorah why would he go somewhere where he couldn’t do that until he turned 30(Average age of semicha there)?
And if someone just wanted to learn all his life, then again, why would he got to a yeshiva that would push him to go into Harbatzos Hatorah at some point?
Therefore, outsiders would not consider going to Chofetz Chaim, and this is also the reason why some of the Rabbeim spoke so openly (within the walls of the yeshiva)about the benefits of their approach. Because within the yeshiva the bachrim would ask those same questions so the Rabbeim had to discuss with them the benefits of the derech as opposed to the others.
(Note: there was a previous where he quoted a talk given by one of the Rosh Yeshiva about Rabbi Shaya Cohen’s speech he gave in which the Rosh Yeshiva tried to explain his words differently. “Counteract and explain” was the quote. I just wanted to clarify so there wouldn’t be confusion that that they were not disagreeing on any major hashkofic issues(as for as i can recall, i am sure you will correct me if i am wrong)but rather on a discussion about weather a certain sefer can be brought to discuss Muchruch Shmoozin(for more about this you can read the introduction to the book “pinnacle of creation” where it discusses the anatomy of the “Shmooze”).
(Also just to clarify I am fairly sure the Rosh Yeshiva quoted was not R’ Henoch Leibowitz but one of the present Roshei Yeshiva who is a chaver to R shaya Cohen and they are quite close.)