This is a classic example of overcorrection as a result of learning language rules at school.
Accusative pronouns have had a tough run in the past 100 years in English. In school, we were always taught to say "Sarah and I are going to the park" instead of "Sarah and me" or "Me and Sarah", and so on. But what often either fails to be conveyed (or is lost on the student) is that this only applies to nominative case; that is, it applies to the subject of the sentence only. Because subjects and objects and case markings are things that people don't normally explicitly think about when making a sentence (your brain pretty much computes this for you), they get it wrong when trying to apply the rule. So, you get people saying "Please see Sarah and I", even though it should be "Sarah and me" because it is the object of the sentence. All of this has had the effect of (in my observation) the accusative pronoun (particularly the first-person singular pronoun) being looked at as somehow informal or uneducated sounding.
Another method of avoiding the accusative pronoun that people have stumbled upon is using the reflexive pronoun, and this is what you're seeing. So, instead of saying "please schedule a meeting with John, Mary, and me" (which is 100% correct), some people perceive it as more educated/formal to say "please schedule a meeting with John, Mary, and myself".