I suggest that first, have the pastor/leadership publicly endorse you as the worship leader (assuming you want that responsibility), or ask them to assist in finding a worship leader if you don't want the role. Then, ask them what they expect of the worship group. Once those two things are in place, the worship leader can "lead", make decisions on who plays, what is expected for members etc. For example, many worship bands expect the players to learn their parts away from the rehearsal period, and come to rehearse the songs and set as a group. As to the level of expertise, arrange the songs within the limits of the group, but encourage them to learn more, and keep looking for "better" players.
Also, ask the leadership to ask for more members during services and in newsletters, etc. Some people don't volunteer to play because they think they need to be invited, or other reasons.
Communication is a key piece for all of this, you write that letting outsiders pick the songs might offend those who do the choosing right now, don't guess, ask them if it is so, they might be happy to share that burden. Communicating the church's vision for the church, and the worship group is important, communication expectations for each group member is important. I once chose not to play in a worship group because their perspective on the role of the group and the music preferred was very different from mine. In recognizing that they were the leaders, not me, I chose not to cause any problems and took a different role.