It depends on exactly what type of a “tropical” vibe you are looking for.
The Polynesian certainly has the most traditional tropical atmosphere, and the pool bars are excellent! However, the most recent re-imagining of the main pool (now the Lava Pool) removed quite a bit of the vegetation and shade. Also, the Lava Pool can be extremely busy, as in overrun with people everywhere. Bustling, busy, noisy. The new Oasis pool is usually much quieter, but probably a lot less interesting to a 6 and 16 year old. The upside is that you could rent a covered patio for shade, privacy and relaxation, and the Oasis bar and grill is right there with easy access. Patio rental is relatively reasonable, too, IIRC in $50-$60 range for a half day (please don’t quote me on that!).
The Poly is a lot more convenient transportation-wise when you want to go to the parks, and with the view of the Magic Kingdom and Seven Seas Lagoon, you really feel like you are at Disney! The Poly overall is my favorite WDW resort.
On the other hand ...
IMO, the more relaxing pool experience can be found at the AKL Jambo. It’s not as in-your-face tropical as the Poly, but still highly exotic. Uzima Springs pool is much larger that either the Lava or Oasis pools at the Poly. It has tons of lush green vegetation and shade, and we never noticed it being overrun like Lava pool. Lots of seating, with a great pool bar and the Mara quick service food nearby. The Kidani pool would also be available to you, but we didn’t visit that one so I can’t comment on it.
The downside of AKL is that the rooms are smaller, with less bathroom space than a Poly studio, which can be a factor for a family of four. Also, you have to take buses to all the parks, no boats, monorail, or Skyliner. But it is also less expensive, and more likely to be discounted, which can save you a bundle. More than enough for lots of Uber/Lyft rides, or car rental/parking if you so choose. The restaurants are superior too, IMO.
Here is a pic of the AKL Uzima pool from our balcony that shows the atmosphere and amount of green vegetation. It is transporting.