> that key combo is hardcoded in LISP

isearch-forward turns on a minor mode that has it's own keymap

a snippet from isearch.el:

  (defvar isearch-mode-map
    ...
    ;; Several non-printing chars change the searching behavior.
    (define-key map "\C-s" 'isearch-repeat-forward)
    (define-key map "\C-r" 'isearch-repeat-backward)

hence to redefine C-s, to, say F12:

  (global-set-key [f12] 'isearch-forward)
  (define-key isearch-mode-map [f12] 'isearch-repeat-forward)

& so on

I don't see what is the problem here