If you want to play the music that will appeal to your own demographic, you'll join a legion of bands that are playing stuff audiences don't really care about anymore, and you won't have much work. There are more ads for non working "southern or classic" rock bands on Craigslist than you can imagine. Some of them are trying to find work, so you're competing with all of that, and vying for an audience that doesn't go out anymore. Full disclosure: I will be 60 in April.
What your band is doing is trying to find a way to give a randomly thrown together audience a little bit of everything, which is not a bad strategy if you're doing weddings, private parties, festivals and other functions where your audience is likely to be diverse in age and background. That's actually a pretty good way to go if you want to be able to charge a premium for your services.
So I'm going to suggest that maybe you look at this from a purely technical viewpoint. If there are songs the singer can't sing, or songs that sound like Alvin and the Chipmunks when they're capo'ed up to the singer's range, those might need to go. But other than that, this seems more like a decision for you to make as to whether you'll be happier doing a classic rock band that probably won't work very much or sticking with this mix that's more likely to be called in for a situation where diverse music is appropriate.
I'm the opposite of you, I think. I am completely sick of bands doing the music I grew up with. But I also don't want to play a lot of newer music, so I play in an originals band that writes really poppy and accessible tunes, and I do a few other odd projects on the side that are purely for fun, playing standards and stuff. I'm not expecting to make much money with it, but it keeps me happy, musically speaking.