I don't get why the enemy has a board full of cheap minions to trade with in these scenarios.
That should be the strength of your deck.
My point is that those types of decks we've seen in the past don't tend to run 6 cost cards. Maybe 1 or 2 at most in the deck since you are so focused on doing the damage to face and winning early. Zoo is the only exception that I can think of, but that deck has been pretty much dead for a good while (RIP PO). All the minion based decks I've seen recently have been aggro and they aren't likely to include a tech card like this because it significantly harms their consistency.