Yeah this would potentially be a nice solution; I do think it’s fair that if you see someone casting a spell you yourself can cast, you’ll recognize what it is. The one drawback I can see is that it requires me to have everyone’s spell list memorized, which increases the chances I miss one of the matches.
The more I think about the arcana check, the more I’m interested in it; initially I was worried adding additional checks every time somebody casts a spell could slow combat down too much, but maybe I’m overthinking that.