I would be comfortable using a QR rear wheel on such a hanger, with caveats:
* Make sure the QR skewer is long enough to engage all the threads in the acorn nut.
* Put the QR lever on the left side of the bike, away from the cassette.
* That silver bolt visible on the left of your image, should go through to a specially shaped nut on the inside which is curved to sit in the top of the dropout. That should be a good snug fit in the dropout and prevent the hanger from sliding or rotating without force.
* Your axle should have just enough stickout on the right-hand side to engage with the hanger but not so much it stops the acorn nut from tightening down completely. Without this, the hanger might rotate a little, throwing shifting off.
Changing to a solid rear axle is a valid solution too. Just remember to pack a 15mm ring-spanner in your on-bike toolkit.