Ben:
A few points to consider.
There are actually two "bathtub" construction methods, the USGA and California method. The main difference is the USGA green has a gravel layer and the California green does not. It's far more accurate to say that the USGA green has a "Hanging" water table and that the California green is the "Perched" water table.
The USGA green "hangs" the water table because the water is retained in the smaller pore space of the USGA-spec sand over the larger pore space of the pea gravel layer. At a critical point, gravity forces the water to overcome capillary action and flush out of the smaller pore space in the sand and into the pea gravel layer.
That being said, it is imperative with both forms of construction to match the compacted subgrade with the desired contours, as pointed out by Jeff. The difficulty with the USGA construction is that there are three layers to consider (Subgrade, gravel, surface) as compared to the California method with just two.
What you are calling native soil greens are probably greens where soil native to the site was available that matched the specifications in terms of particle size of the USGA or California constructions. The specs allow for the desired resistance to compaction, as well as the desired drainage performance and other characteristics. Any sort of modifications outside of the USGA parameters and the green will not perform as advertised.
Your question is best answered by looking up the exact USGA specs in terms of root zone mix, and amending the soil to match those parameters.
Remember, the chief motivation behind the development of the USGA construction method was not drainage, but resistance to compaction.