New Homeowner/Pool Owner

Bought a house in October for the first time; it came with a pool that the previous stated hadn't been serviced in 2 years - cranky neighbor says 4.

28' Round is about all I know; Hayward EC40 filter. We left it as-is over the winter, with a cover that was falling apart. Bought a new Winter Cover. Got sick of hearing the frogs, so now I'm cleaning it up. Removed cover, which deposited leaves and mud into the pool. Drained it (It's Above-Ground - I read the warnings). Patched several holes where tall reeds poked through the vinyl. It's now mostly drained and cleared, except for two 1/8" puddles of mud. I have taken apart the filter, and am currently soaking the fingers in DE cleaner.

I think my plan is to re-fill the pool, using shock or chlorine tablets as throw-away to keep the frogs and mosquitoes at bay. Then, disconnect my DE filter from the pump, and use the backwash hose attached directly to the pump to just vacuum to the yard to get the last bits of mud out.

Does anyone know if those fine meshed leaf eater things work with mud? They're pretty fine particles. The previous owner has a couple vacuum attachments, one of those triangle-shaped ones for sure.

Once the dirt is mostly gone, I then plan to refill the pool as needed, attach the DE Filter, and add the DE. Then I can start worrying about chemical balance, and have a working pool - right???

Trying to do all this on a budget, too, to boot...