Sorry I missed this thread somehow. I would definitely use the trackhoe to remove beaver lodge debris. That will set the stage for a dozer to come in and rearrange 'good' soils and rebuild the levee. Plus, with all evidence of the lodge gone this time of year, any beavers left will likely move on. If not, they will be obvious and you can take action. Looking at your photos, there doesn't seem to be any new beaver activity, except maybe at the right in photo 4. Lodges look like they are in disarray.
The concrete structure looks like an old cistern, used years ago to collect rainwater off rooftops for household use. There may have been a homestead there years ago.

Teach a man to grow fish...
He can teach to catch fish...