Yet again, the devs messed up what could be a decent brawl mechanic. By turn 3 or 4, your hand is full for the rest of the match, pretty much guaranteeing that you won't get to play the card you chose.
The opponent's hand is also likely full, so they don't receive the cast off card.

Best way to have avoided this would have been to set player's opening hand to zero cards or expand hand size to 12 or 15. With three cards being added to your hand every turn, you will still eventually reach a point where your hand is perpetually filled, but at least it would take a bit longer and you could actually play the cards you picked and/or received.