Charles from CA made an interesting comment on a thread about the 80's, essentially comparing the "Glory & Praise" music to the hymns of the "St. Gregory Hymnal", as both are substandard music which became introduced into the Mass first through popular devotion. (That may not have been his intent, and I'm not quoting him verbatim, so don't attribute those actual views to him) This is the history of so much bad music and bad liturgy. Suddenly, the answer was obvious:

Promote quality devotions.

I'll give a few examples: at my former church, many devout people prayed the Divine Mercy Chaplet along with a CD of a pop arrangement of the litany. How could they embrace good music in the liturgy when they listened to pop music in church for a very important devotion? If your church has a sizable number of people reciting the Chaplet, you might consider seeing if they'd be interested in chanting it.

Although not devotional, I started a concert series at my current (Episcopal) church, and the fruits of respect for music by the congregation have been astounding. Hold concerts at your church. (If you have reservations about holding it in the church building, use a hall) Concerts of secular classical music will do wonders.

Play or sing for Stations in Lent. Starting a Vespers would be beneficial too. Sung rosary. Follow "Bring Flowers of the Rarest" at the May Crowning with "Regina Coeli". Just a few suggestions, but promote quality musical and/or Latin devotions, and it will seep into the liturgy.

And let me say that it would be spiritually toxic to do away with more "folksy" devotions. You aren't "fixing the music" at them, or giving them something "better", you're taking away their prayer life. I'm saying offer higher quality options as well.