arlie I think you are correct on all counts.
You asked about leaving some behind to seed the microbes into the next batch. I would use fresh compost each time. What you are getting with the compost is a fresh supply of microbes. Some of those microbes cannot survive the water "torture." If you continue to use only the survivors from each batch, I'd be afraid of not having much of any benefit after a few cycles.
Something we never talk about is how, or why, to clean the equipment. The equipment should be cleaned after every batch. There is an unseemly dark side to compost tea. The bacterial always leave behind something called biofilm. This is the bacterial slime
you feel on the equipment afterward. It is also the stuff that grows on your teeth where gum disease forms. It forms underneath the biofilm. Fortunately fresh biofilms are easily brushed away with a toothbrush or a bristle brush in your compost tea maker. I would supplement the cleaning brush with a 1/4 cup of OxyClean or the Walmart version called Sun Oxygen Cleaner. Those products generate hydrogen peroxide to sterilize the equipment.