How to get rid of Rocky dirt?
There are machines which can sieve them out. Comes in modest to large capacities (here is a UK example) https://www.screwfix.com/p/scheppach-rs400-automatic-rotary-soil-sieve/7330k
a handful of rocks a day....what other way?
The fast way would be to hire a Bobcat and dump truck and dig up as much of the old rocky dirt as you can and haul it away and bring in fresh topsoil and reseed a lawn. Or just topsoil over the rocky dirt and reseed. The slower way would be to sieve/sift the dirt a section at a time to remove all the smaller stones and pebbles. Rig up a 1 X 4 wood box frame perhaps 3' X 5' and staple some 1/4" hardware cloth to one edge to make a sieve ...then dig up a section/layer of dirt and sprinkle it into the frame/box and shake the box to sort out the pebbles/rocks and drop the clean dirt below the sieve. Make sure to do this when the ground/soil is dry. Periodically remove the buildup of stones/pebbles from the sieve and shovel the sifted dirt back into the lawn area that has had the stones/ rocks removed. With 6000sf, it will take a long time to sieve it all but I've done that method on smaller areas (800sf) with good results ....just takes patience and time. Spread the work over a couple/few years if need be. https://www.youtube.com/watch?v=ADwlfXC4YjQ https://www.youtube.com/watch?v=ioXGL-5hhT0
other than replacing the soil, there's not much more you can do other than rake, rake, rake