I’m not using use-package, therefore I can only guess.
I can’t see here that you associate flycheck-mode to elixir-mode. So unless you do global-flycheck-mode anywere, you need to (add-hook 'elixir-mode-hook flycheck-mode) at a place where it is appropriate, maybe use-package has some special syntax for it.
PS: Even though I generally prefer emacs over everything else, I currently use VScode for elixir, because language server support is so much better. I hope LS support will be available in alchemist.el soon!
