Immutable here just means there is a base OS+libs that you don't touch. So now elixir or an ide would install in a sandbox with any needed libraries not included in that base instead of install all the libraries and stuff globally
if mix can work without sudo|root it will absolutely work in an "immutable distro" on the other hand: this particular immutable distro may not have all the c libraries BEAM/Elixir expect in that base and while SilverBlue does let you add to the base this doesn't sound like it will. So it might take some effort, hard to say at this point, though you can always add to your PATH
