It sounds like a good idea, and no it will not damage root systems as long as you do not get any on the vine. Don't spray the trunk of the vine, or any leaves. Watch for drift!
What you state should work and work well.

My property overlooks two vineyards with different growing methods. The first is both organic and bio-diverse using no chemicals period. He has a machine tractor mounted he calls a French Hoe which is able to machine hoe between the wines. Second vineyard uses the most popular method here in the Napa Valley wine country of spraying with a commercial formulation of Glyphosate, completely machine sprayed. Both vineyards plant a cover crop between the rows which is tilled in later in the season. Al

Spreading newspaper several layers thick under mulch tends to keep the grass and weeds down. I've not tried it around grape vines, but it has worked excellently on my blueberries. I am surprised to hear they are using Roundup on grapes in California. I had read that grape vines were extremely sensitive to it and not to plant grapes within 1/4 mile of fields where drift could occur. The railroad uses Roundup on the right of way, so I avoided planting grapes along that side of our property. Maybe I should change my thinking on that. Northwoodswis