I suspect it would mess up the system. The rings need to be able to "lock" together and stay reasonably aligned. Pulling on them with two separate ends of webbing isn't likely to do that very well.
But you can try it out and see. If you don't want to sew them in just tie them on with a larkshead or girth hitch to see if it works.
The whole point of the using a biner is so you don't have to thread the one end through. If you're going to do that just use a sewn loop on the end. No need for a ring.