Initially, I liked this idea, but a bit of thought makes me very much dislike it. Kudos, still, for out-of-box thinking.
Here's what turns me off. You have three basic scenarios: 1) win-win, 2) win-lose or lose-win, 3) lose-lose.
In cases #1 and #3, it doesn't matter what the control die is. Everyone knows the outcome. There's no suspense other than, maybe, trying to delay the lose-lose to the point it doesn't get you killed. May as well go with a passive check.
For case #2, assuming the DM and player dice have equal odds of being "right", you've reduced another probability to a coin flip. Unless the DC was already 10 (11, actually), in which case you've gained nothing. If the odds aren't even, then you're still just replacing one probability with another, which could swing in either direction.
The odds of #2 being relevant depend on the original DC, anyway. If the DC is 20, then you've only got a 9.5% chance of it mattering (either die comes up 20, while the other doesn't) and converting a 5% unknown to a 50% unknown. FWIW, a 15 (or 5) DC has a 37.5% chance of mattering.
Sorry. I like statistics.