I use washable school glue and either let the binding dry naturally (while I do something else) or hit them with a warm iron to dry the glue. For bindings I find the glue stick works best. I have a friend who makes skinny bindings and I agree with you, they look so neat. Try starting off with 2" strips.
Although, you could make the backing larger, fold over and press 1/4" and then fold over to front of your placemats and straight stitch or zigzag in place. That is what my sewing group at church does on the quilts we make for charity.