It's good to know you're not giving up on these beautiful placemats! The main advantage to doing the binding separately is that it will be doubled and will hold up longer. Since placemats need to be washed frequently, that might be especially important. Also, I think it's often easier to just do the separate binding that to do the planning and trimming that is needed for the other method. I've done them both ways, and that's just my opinion. It doesn't save fabric to fold the back over because you have to allow extra in case it pulls in too much during quilting. There is an excellent binding tutorial on Youtube. Search there for "Binding the Angel" by Sharon Schamber.
Welcome to QB! I hope you enjoy quilting for many years to come.