There are many existing cards that aren't fully active without other classes of cards in the kingdom, e.g. Moat, Courtier, Tunnel, Fortress, Hunting Grounds, Mission, Labyrinth, Changeling, Patron, Falconer.
Considering full random can be useful because it gives a metric for whether a card is too narrow. If the circumstances are too rare, then most of the hand-picked games with that card will feel the same. I know that's how I feel when I try to make Crossroads really effective.
I think that both random and constructed games must be considered because both are played, but not all cards are required to appeal to both play styles.
Also, I hesitate to say this because I don't want to taint peoples design choices, but self synergy is not required for my challenge (and neither is it discouraged). The stipulation was that the card can't have trashing itself be the benefit. The card could trash other copies, or it could just not have inherent trashing ability. Catapult/Rocks is acceptable; so is Catacombs.