In `ido.el`, the `ido-everywhere` minor mode overrides
`read-file-name-function` and `read-buffer-function`. However, it does
*not* override
`completing-read-function` with `ido-completing-read`.

Should it? This is what Helm
<https://github.com/emacs-helm/helm/blob/v2.6.1/helm-mode.el#L1247> does,
and it is convenient for the user.