As for web hits for *'amok run,' compared to the terms that are really used in English,
"(run OR runs OR ran OR running) amok" - 1,250,000
+"a rampage" - 1,030,000
I would say those relatively few hits are negliglible enough to disregard. The vast majority of them look frightfully Denglish to me (even some websites that don't end in .de); in fact, many of them may have been caused by this very entry in LEO, which is obviously wrong, but obviously not so obvious to German speakers.
Google is not the measure of all things; the English speakers are unanimous that 'amok run' is not real English, and no one has found a citation for it in any English dictionary. So there's no question that 'amok run' should be removed, or that 'rampage' is a suitable translation.
The only question is whether to include an additional translation or two such as 'killing spree' or 'killing frenzy' ('crazed action' seems more like an explanation than a translation to my ears) that apply only in certain contexts. I don't have anything against extra translations, as long as they mean the same thing (or a subset of the same thing) -- it's just that 'carnage' doesn't.
'To go postal [coll.]' could also be a useful addition, if it hasn't already been suggested elsewhere (there should be several threads on it in the archive, but I don't remember if they were in New Entry), but it's not a noun, so not as helpful in solving the question at hand.
I do agree that this change should be a high priority, since the word is so much in the news this week. IMO it would be better to go ahead and change it to 'rampage' immediately and discuss additional options at our leisure than to waffle around and continue misleading people.