Mostly only the bent-gate (or biner that will be clipped into the rope) have these 'o-rings' or 'Strings' (what Petzl call them).
The reason that your straight gate biner (the one you will clip into the bolt) is normally not fastened is because it generally does not turn upside down as it is weighted by the lower biner while hanging on your gear loop. I'm sure price does come into it, but it’s also convenient to be able to break up your quick draw for many reasons. For Trad climbing is one reason.
Primarily though, if you've dogged up a route but would like to make a Red point attempt, you pull the rope down (leaving your quickdraws on the route). If your rope clipping biners (bent gates) are not fastened with a String (or other) there is a good chance your bent gate may turn, making clips a little trickier while holding the crux crimps during your Red point attempt.
Should the biner clipped into the bolt turn, no big deal as you can easily get to it while being lowered off the route.
The Petzl Strings weigh 2 grams each.
Just don’t do this with your quickdraw