Yes I would put a filter on the well pump to keep the dirt out of the cistern. The foot valve leaking is why you are losing prime. Remove the above ground check valve and the pressure tank will keep the pump primed. However, if you don’t fix the foot valve, the pump will then cycle back on when you are not using water to make up for the water lost through the leaking foot valve.
You can lay a submersible on its side, then you won’t loose the bottom 12” of tank capacity. However, keep the sub off the bottom a little so it doesn’t pick up dirt. See the picture. This is a good way to lay a pump in a cistern, but the pump needs a shroud or flow inducer sleeve.
And yes make a lid of plywood or sheet metal so you don’t have to dig to the cistern any more.