Your deduction, that not everybody's cup of tea means not to everyone's liking, is absolutely correct. Joe got everyone's attention and started to speak. There is a comprehensive article on the topic on grammar girl:
Are the words everyone and everybody singular or plural? May i have everyone's attention,. Unless of course you’re a buddhist or a cat, in which case everyone has multiple lives and both would be correct, depending on context.
There are actually two issues concerning this topic: The possessive form is everyone's (the form everyones' is incorrect).example: An example would be it flew over everyone's heads, or it flew over everyone's head. The possessive of everyone is everyone's, in the same way the possessive of everybody is everybody's.
Should it be everyone's, everyones' or everyones? What would be correct in this case and why? The expression commonly used in both the affirmative and negative. In fact, to catch someone's eye is an idiom, not eyes. the question is not about everyone being singular or plural.
Everyone who is coming will receive a gift. Would the noun following everyone's be plural? In my experience, the spelling checkers get confused by contractions. The indefinite pronoun 'everyone' is a singular pronoun.