Do I understand this right?

[Leefrogs]

I would say whites need a filter as they like to poo in water, and everywhere else. Bigger frog, bigger poop. They're dirty frogs and I couldn't see doing water pool without a filter.

[DonLisk]

I'm not a fan of a under gravel filter system.
One problem with an under gravel filter (unless your covering it with gravel) is that it will slowly over the years clog up unless the water is being filtered before it gets to the under gravel filter. If it eventually has a problem, you must be able to access this and ripping out the tanks substrate is not a fun job.
You might as well call it a re-build.

A pump that is accessible with a filter on it will be much easier to access and clean if it does become clogged.

The false bottom is mainly a place that water will flow to as it makes its way through the substrate to allow the substrate to drain and not mold or rot. The frog doesn't have access to a false bottom or at least shouldn't have.
The water there will be filtered by the substrate or through the gravel in the pond area as is soaks through the bottom of this area and drip into the false bottom.

If the frogs poop in the pool, the waste product will stay in the pool and not move on to the false bottom.
You will be more likely cleaning the pool area of the frogs waste.

Here is a link to Mikes images and the false bottom he did. You'll see how he created the water area with stone and it will drain off to the rest of the false bottom.
Frog Forum - MikeM670's Album: 55 Gallon Vivarium Build