How can I more easily see interactively whether `overwrite-mode` is enabled?

If it is the position of the key that is the problem, then you can unbind the key so that it does not enable overwrite-mode any longer:

(global-set-key (kbd "<insert>") nil)

If you still want to be able to enable the mode with a key, you can bind some other key to it, perhaps a function key:

(global-set-key (kbd "<f8>") #'overwrite-mode)

although you will have to figure out what would work for you.

In the end, I used https://www.emacswiki.org/emacs/AlarmBell and added some code to overwrite-mode-hook. So now I get a visible alarm, which may be better than completely disabling the key. Time will tell if this is better than the other answer.

(defun double-flash-mode-line ()
    (let ((flash-sec (/ 1.0 20)))
      (invert-face 'mode-line)
      (run-with-timer flash-sec nil #'invert-face 'mode-line)
      (run-with-timer (* 2 flash-sec) nil #'invert-face 'mode-line)
      (run-with-timer (* 3 flash-sec) nil #'invert-face 'mode-line)))


(add-hook 'overwrite-mode-hook #'(lambda () (double-flash-mode-line)))