It's too rocky here to ever be neat and tidy and we avoid manicured lawns. We hunt tot lots and in the woods where you can just kick it back and level it with your foot.
For lawns, I've read before and it makes sense, if you cut only three sides, with the fourth side serving as a hinge, you can lift up the sod as a flap. That leaves some roots connected and it will recover more easily. Fold the flap over and you have access for more invasive digging.
Maybe the bulb planter could be modified to have a wide gap on one side to leave some roots. I've tried this on a lawn once and it works well enough, but I never went back to see if the grass benefitted from it. As I say, it sure makes sense when you consider how grass grows mostly from rhizomes and travelling, intertwined roots just under the soil's surface.
Some people carry an old kerchief to lay the dirt on. Pour the dirt back in after digging and replace the flap. If the soil is dry, give it a drink from your canteen and move on. But not too much water, or it makes a green spot in an otherwise dry lawn, which is just as noticeable as a dead spot in a healthy lawn!
A little practice with these techniques on your own lawn should give you good results elsewhere and a chance to observe what happens to your plugs over time.
-Ed