In a forest pool, there is almost certainly some "muck". This is decaying organic matter - probably leaves and twigs for your pool. You may also have inorganic sediment (soil particles) that has washed into the pool. This is usually called "silt", although that is not always technically correct. The silt would also feel squishy to walk on.
I am still not clear, do you have a lower elevation of land available to drain water from your forest pool? If so, then I believe your project should work, and probably provide some relief.
The muck and silt in your pool probably does provide a poor seal. So yes, a french drain on top of the pond muck layer would convey away surface water. However, if it is truly a ground water pond, then getting below the muck and silt would allow you much more efficient drainage of the surrounding aquifer.
Can you dig down a little ways at the edge of the pond? The muck should be a little thinner there before you encounter your true subsoil. If possible, dig a foot or two below the elevation of the middle of the pond to test your deepest layers. If you only encounter clay, then probably NOT a ground water pond.
Just breaching the side wall deeply enough and letting the pond contents run down the slope could be sufficient to drain some ground water away, especially if you scrape some muck away so more of the sand/gravel layer is exposed.
