Here's a few rough concept illustrations of my gravity flow filter system to twirl around in your brain while you set fence posts.
Filter vessel water levels are at same level as pond, thus the 'gravity' flow, pump pulls water thru all vessels and returns it 'clean' to the pond.
Valves(not shown) placed between vessels, and in vessel drains, gives control for dumping debris collected from pond.
This would work with the 'bottom drain' you installed in your pond.
I have 'retro drains' in my pond instead of bottom drains.
This shows more detail of contents of vessels.
90 deg inlet in first vessel creates vortex that helps heavy detritus to drop out of water column, drain needed in this vessel(not shown here).
Second vessel has graduated sheets of different coarseness of filter material.
There are lots of way to configure the mechanical and bio filtration aspects of these systems,
but the basic theories of a vortex settlement first in line and the pump last in line are essential.
Size of pond, piping, pump and vessels need to be balanced.
More later if you're interested.