Removing a significant amount of the nutrient laden muck should eliminate the duckweed problem. Going forward you might see a lot more filamentous string algae.
Bottom diffused aeration will definitely shorten the turnover rate to easily once a day. One dual head diffuser with air 2.5-4cfm from a 1/4hp air compressor (rotary or rocking piston). should be able to create a water turnover in 3 to 5 hours. Depending on shape of the pond you might be able to get adequate pond mixing by running the pump with 2 diffusers two hrs per day.

I suggest definitely wait until the pond clears up to 2.5 to 4 ft of visibility before running / installing an aerator. If you start the aerator with very fine silt still in suspension the aerator mixing will tend to keep the smallest silt particles in suspension. If the pond gets some ice cover this helps clear the water. Generally the water is clearest in a pond just after ice out due to the prior period of no wind action and very slow biological activity during cold water of winter.