I am a Swede and in our alphabet we have three "extra" vowels -- å, ä, ö.
These are in fact short forms for aa, ae and oe, a 16th century printing invention where the second letter was put on top of the first, and made smaller. (This still prevails to some extent in hand-written texts, where the dots are instead a squiggle to imitate the e.) In Danish and Norwegian, another solution was implemented instead of the diacritical dots, that of either writing the two letters very close or on overlapping. In Swedish we don't speak about diacritics, these three are completely different letters with specific pronounciation attached to them. (In Finnish it is extremely important because the spelling and pronounciation are completely in unison. I can read out a text loud to a Finn who will understand it all even if I haven't a clue of what I have said!) Thus, we cannot omit the diacritics and only write a and o, because this would change the pronounciation and the meaning. We have nine different vowels to use and form the sounds that we need, and there is on the other hand no need whatsoever to make vowel combinations like in English, combination that do not even tell you how to pronounce them. Therefore, it is also very annoying (in a way) that our webworld is so devoid of these diacritics; omitting these "extras" makes our words look ridiculous at best (and provoking at worst). You cannot say that an R is a P with a diacritic (an extra leg added), can you?

Language structure has probably decided. We should, I think, in the Scandinavian context consider how words are related, for instance how you make a plural of a noun. In English it's very simple, some kind of pigeon style (because of all the peoples having invaded Albion). In Swedish there are six different so-called declinations, five of which using suffices. In this process, Umlaut may also occur (as in German). I think that this is a very strong reason why diacritics are not needed in English.
And, oh yes, a Welshman at our job insists that there should be a diacritic sign in "role", whatever role it may play; at least it does not change the pronounciation.