We have a diacritic in English text already. Rather than above or below, it goes to the right of the letter it modifies and looks an awful lot like a letter h.
And if you don't quite buy that, remember that a lot of diacritics started life as letters that were eventually moved above a preceding letter and then simplified. The tilde on ñ was an n itself; the ring on å was another a; and in at least some cases the umlaut was an e.
Modifying-h may only be stuck where it is because technology did away with the need for economical scribes before they had a chance to start messing with it.