After that, you'd need either new letters or a lot of diacritics if you want to avoid multiple readings of the same word. The letter "a" can be pronounced something like three or four different ways, and clearing that up would cause written English to resemble Pinyin (or maybe Croatian). When choosing the new writing system, try to keep existing computer support in mind or nobody will be able to backport your new system to ASCII-only text.
There are many many reforms that can be done to English spelling that would simplify it tremendously without making it fully phonetic. Apart from reversing some of the silly forced latinization, you could get rid of some of the more bizarre letter combinations, like "ough" (could be replaced with, say, ow/off based on word). And sure, even here, there may be some dialectical differences in some words, but they would be rare. No one pronounces "though" like "thoff", so why not write it "thow"?
this may have been a case for their introduction in the first place
You would not. You could instead make the rather sensible choice of allowing the written forms of distinct, increasingly divergent, dialects to be different from one another. There are several clusters of similar but differing written standards existing side-by-side in Europe, and they don't pose any major challenges to communication. And in some language regions, it's quite common for people to write in their local dialects in informal contexts instead of the standard variety, which also doesn't pose any major challenges to communication. There is very little actual necessity for a single unified standard variety. By the time the dialects would have diverged far enough for there to be issues in communicating due to distinct written forms, perhaps a few centuries from now for English, the alternative of having a unified written form would necessarily mean that to be literate at all, you'd have to learn a foreign language (the unified standard), which would be the worse situation to find oneself in, and it would probably be easier to first acquire literacy in your native language, and then learn the standard as a second language.