Apple Music in commercial setting

Hi there! I represent a non-profit, international company. My specific department deals with productions, concerts and conferences in multiple, sometimes mobile locations. We typically need some sort of background/walk in music for these events, but continually struggle with the difficulty of: (A.) making sure we are paying the correct licenses to play the music and (B.) having a sort of central music library for all of our different staff to pull from.

The easiest solution in my mind is as follows:

Simply pay for the proper licensing (ascap, bmi, sesac etc) and use a service like Apple music to create playlists that would be accessible to all of my staff around different properties. This allows me to make one easy to update playlist without individually loading up dozens of mp3 devices or sending out countless CD's.

My question is as follows:

Is this legal for me to do? I find Apples terms & conditions on this a bit confusing... I understand the service only allows you to use it in a private, non commercial setting and I imagine the company simply does not want to be sued because of people paying for their service thinking they now have rights to use it publicly. All that said, however, if we do indeed have the rights to play the song publicly, however, does that mean I am in the clear to use my music player of choice (a.k.a Apple Music).