French drain dries out a wet area, it will never handle the volume you are dealing with. Correct me if I am wrong, it seems the water comes off the pool deck, from the yard by the pool, and the little area between the house and the pool deck.
I would do 2 things. On the left of the pool make a swell so the water does not flow towards the pool but is directed to the woods.
Next install a channel drain against the pool deck. This drain should be on the house side and the left side. The have a 6 inch pipe coming out of this dumping in the woods. You can buy sections that will need to be incased in concrete. You can also build you own. both require lots of digging and concrete. The DIY can be deeper than the prefab to handle the water. You can buy grates for the cover in plastic, fiberglass, aluminum, or steel.
I would do 2 things. On the left of the pool make a swell so the water does not flow towards the pool but is directed to the woods.
Next install a channel drain against the pool deck. This drain should be on the house side and the left side. The have a 6 inch pipe coming out of this dumping in the woods. You can buy sections that will need to be incased in concrete. You can also build you own. both require lots of digging and concrete. The DIY can be deeper than the prefab to handle the water. You can buy grates for the cover in plastic, fiberglass, aluminum, or steel.