That'd be super thematic. We've actually tried a lot of different angles on caring about your opponent's stuff like this (not sure if we've tried this specifically). In my experience working on various games, it usually comes down to the same issue:
When your opponents have keywords on their board, it'll feel great. When they don't, it'll feel frustrating. Matchups also become a lot more polarizing, with some matchups just not letting your character do much of anything.
For a point of comparison, imagine you see the following two quests were proposed for an event by a designer:
Quest A: Summon 30 Shurima units.
Quest B: Kill 10 Shurima enemies in ranked mode.
Quest A asks you to play a deck with Shurima units. Very doable. You can play a shurima-focused deck and do it fast, or you could play a deck with a few shurima cards and do it a bit more slowly. Either way, you can reliably progress your quest.
Quest B will only progress if you happen to matchmake against an enemy playing Shurima. You don't control what your opponent is playing, whether your quest can progress or not is totally up to the whims of fate.
Cards that are extra-good if your opponent is running a certain strategy have their place as meta answers, but making the core fun of a champion only work if your opponent is playing certain types of cards is a lot more frustrating for many players.
There are surely exceptions where this could work really well of course, just because its tricky doesn't mean its never a good idea. It depends a lot on how common the thing is that your deck requires your opponent to be playing For example, you can assume most decks will be summoning units at some point or other so cards that only function if your opponent has a unit (like removal) are pretty safe to include.