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as the chair of an australian public broadcaster's english usage committee, i can unreservedly say that americans are free to pronounce "emu" with or without a palatal consonant: /ˈimju/ or /ˈimu/ https://twitter.com/stu_rush/status/1296820294036922368
if you are interested in the historical process of sound change undergirding this and not just reinforcing your own normative opinions on language use, please read on
historically speaking, some words ending in that /u/ vowel in english (e.g. "blue", "rude") were once pronounced with a palatal — /rjud, blju/ — but are now not. this process is known is yod dropping
yod dropping is common in many american varieties of english (e.g. news, tune, tutor). but it is also variable in AU/UK varieties (e.g. in words like "assume"). and total in east anglia, where apparently "beauty" and "booty" are homophones
i am no expert on phonetics or US speech , but seems to me that if "music" qualifies for yod-dropping in your variety (i.e. MOO-zik) then "emu" surely also could
if you are interested in reading more about the fascinating history of yod dropping, here is a great article to start with http://www.cercles.com/n22/glain.pdf
just jumping in to fix an upthread error: "music" is not yodless in GeAm, though nasal consonants (e.g. news, nuclear) are a prime context for yod dropping
