Well, the Arabic script is an impure abjad in that it does write long vowels so that lessens the problem, but the ambiguity sometimes does come up, at least according to Wikipedia.

It probably also just means that Persian is better suited than English. I would think that due to the English wordstock being full of loanwords from different language groups that you'd have more collisions than you otherwise would. English also has an unusually large number of vowel sounds, which makes it plausible that a lot of our words are distinguished by vowels rather than consonants.

Also, I wouldn't rule out that writing English in an abjad only seems bad because we aren't used to it, kind of like judging an alternate orthography because it looks weird, but my hunch is that we'd run into problems. That's actually what happened with the Greeks when they started to write initially by adopting the Phoenician script, which is an abjad. They realized it was too ambiguous and added letters to represent vowels.