That's just bad design. The pool builder is on drugs if he thinks another catchbasin in the same overloaded drain line and some grasses will fix this. As you identified, the issue is that all the water from the woods uphill is doing what water does and finding the low spot. I would have liked to see a taller wall, high enough that a swale could be built behind the top of the wall to divert the water around the pool. Depending on how much space you have you *might* be able to do a low wall in front of the fence to accomplish this. This is going to require smart grading and probably additional hardscape work. I don't see french drains and box drains fixing this.

Revolutionary Gardens is correct.

The back fill on the wall must be gravel with a perforated pipe to wick water away.

While the planting bed looks nice adjacent to the raised area, is not practical. Even minor rain washes mulch and dirt into a pool.

Yes, plants on the upper part of the slope will help, but young plants will have to mature to be effective and right now you need this handled immediately.

Dismantle that wall and rebuild taller, proper backfill and perforated pipe. If your area experiences freeze/thaw conditions, winter rains that freeze up will eventually topple that wall from pressure from behind.

If your pool is concrete, I hope the plaster finish survives. I've seen the very same thing happen on a two day old plaster finish, that was destroyed.