I wouldn't expect that patch of dirt to grow grass back with just fertilizer or treatments.
This is the grass seed I use, Athletic Mix from Steve Regan in Murray. I'll have to take some pictures later this morning but after mowing my front yard last week real short I identified the low spots and added sand and some topsoil to try and raise them up. This was where I had to dig after putting down sod last year which made the spots sink or where there were some gaps in the sod. I bought a 5 lb bag of that seed and plan to put some down on those spots and also generally overseed the whole lawn as well.
I'd aerate your whole lawn then but a 25 lb bag of seed and spread that out adding some extra on the bare spots and if you felt like it add a little top soil to the bare spots as well. After you do that though you gotta water them to make sure it stays damp long enough for the seed to take. I shared some pics in that other lawn care thread from a few years ago on the progress of an area I leveled out and seeded from bare dirt to thick grass.
What I've found is that yard work becomes overwhelming very quickly if you do nothing all year then decide to clean up the whole yard. I try and budget myself 30-60 minutes on top of what it takes to mow and trim the lawn each week. In that time I'll pull weeds in the gardens, trim trees, or pull rocks. If you can keep to that schedule after things are under control then having a decent yard is manageable.