Allow me to chime in with a different opinion. I've considered this for a while, as one who either leads or plays sax. Would it not be great to do both simultaneously?
I decided not.
IMO, the idol issue is a big one. Trying to do two things at once, assuming there are others that can also fill either role, speaks to me of an enhanced sense of self importance. Of course that my be different if there is really a need for extra hands on deck! I would find it hard I think to not be distracted, wondering I guess why this leader felt the need to do everything.
Song or worship leaders in my experience do just that - lead - and in our sound mix their voice is prominent, the one that the congregation listen to for cues and direction. Perhaps if the mix is different and the choir is huge that may not be necessary.
Practically, I would find it cumbersome to juggle a handheld mike, sax with clip on, or another floor standing mike, cue cards etc. all whilst trying to focus on leading the service. I think my multitasking is perhaps limited.
I personally would prefer to lead the service, direct the musicians, singers and congregation as well as focus on being sensitive to what is happening in the congregation, OR, focus on the music, the other band members, and doing the best I could with each song.
Personaly criticism not intended at all. I am fascinated that people would want to take it all on at once. Frankly, I am in awe!
For a worship leader to delegate the leader function to another singer for a particular song that 'needed' a sax part, and then inconspicuously slip back into the leader role would be fine for me. But to try to be a musical soloist, vocal leader, and director all at once: definitely not for me!