I don't think "immutable distribution" typically means "can't install applications", it's more about the system files than anything, not across absolutely everything, similar to "functional programming" doesn't mean "no side-effects allow anywhere" because then you couldn't draw to the screen. All those OSes have included utilities for installing packages ("programs"), otherwise they wouldn't be very useful.
Besides that, even going by your own understanding, if you install homebrew on a immutable distribution, doesn't that mean homebrew is "antithesis to the concept" too, as much as any other package/program manager?