usres -
ofcourse arabic existed before islam and ofcourse all words come from some root or other; so this is nothing unique to the word muslim.
The normal meaning of the term muslim is a person who follows islam. it is used synonimously with muhammadans, mohammedans, moslems etc. right or wrong lingusitically, that is the general usage.
Dictionary.com defines muslimas an adherent of Islam and as “of or pertaining to religion law and civilization of Islam”.
so when you say muslims existed before prophet Mohammed and Islam came about, it is incorrect going by the broadly understood meaning of the term muslim.
It also wrong in a more important respect which is that the jews and christians do not accept being referred to as muslims.