Do you happen to know the full "check to spawn" area? I read 17x17x9?
This area is a square/cube, centered on the spawner. So from the spawner, 8 blocks out on the x and y, and 4 blocks up and down. Mostly this is looking for how many mobs the spawner has already spawned, with some wiggle room, but it's really not very big. If it finds 6 or more (or more than 6 - not sure which), it skips the round. Each round spawns 4 at most.
What's important for efficiency is moving the spawner's mobs *out* of that range as quickly as possible, so more can spawn every round. Flowing water to a single pathway out, like you have, works great. The most interesting way I've seen was water that quickly pushes out, and drops them 5+ blocks down with a water breaker.
Note that the 8x8x3 spawn area is also a square, centered on one corner of the spawner -- most people do a 9x9x3 space for simplicity. The distance you can stand from the spawner and get spawns is 15 blocks away, but the space is a sphere (which makes diagonal distances tricky).
Mobs that spawn naturally in the world have a cap, a max amount that can be in the world at a time. Mobs that come from spawners do not contribute to this cap, and do not have a cap of their own.