That's a good way to clean antique or fragile quilts, except that you don't want to agitate them.
For modern quilts, washing them in a front-loading washing machine works just fine. I don't have one of those, so I go to the laundromat. You can dry them completely or partially in the dryer, or you can use the laundromat's spinning machine to remove excess water weight and then hang it or dry it flat.
I use regular laundry detergent (I make my own laundry detergent) and no fabric softener.