I'd get the soil test but I gonna go out on a limb and predict the % of organic matter is going to come back very low. You'll want to see %'s in the 4%-6% range and that's not easily done in a short period of time. Aerate and top dress is a great plan but you'll also need to mulch clippings back into the soil, leaves too. Fertilize with a product that will stimulate the microbes in the soil. Restore is a good product but there are others out there. Look for something that has a grain meal as part of the ingredient list. Soymeal, cornmeal, or even Alfalfameal. But be fully aware that it will take time. You can go the Scotts route and you'll get much faster results but the chemicals do nothing to address the fact that you have crappy soil. Chems don't build soil fertility so if you make that choice understand that you'll be using chemicals to fertilize and deal with weeds and pest/disease issues for as long as you own the property.

rb33 brings up a good point about leaf bags on the side of the road, but I want to share my experience with this after my former roommate did the same thing for a season or two. basically, the leaf-grabbing, leaf-mulching deal had interesting side effects, some great, some terrible. after a winter had come and gone and seeds had time to stratify in some cases, we found all kinds of interesting new plants coming up around the yard. NONE were native, but some were very welcome additions. of particular note, we wound up with about a dozen Nandina bushes that i gladly hacked to bits after reaching suitable size. we ended up with three unknown Japanese Maple trees. it's been about 14 months since they sprouted, one is 6" tall, another is 12" tall, and the last one is almost 18" tall at this point. i transplanted the smaller two into pots this weekend, and my rear deck's Oleander is looking rough right now, like it might croak, so the third may go in its gigantic pot. finally, the one other thing that came up (which reseeds profusely) is purple Shiso. i transplanted it as well, and we still find little sprouts popping up here and there and everywhere - i have about two thousand seeds from just one plant. anyway, just sharing the fun things that can happen when you grab bags off the street. be picky about the houses you take them from. if they have nasty invasives or lots of weeds, your 'free blessing' may end up a 'nasty curse' quick, depending on what seeds got swept in with the leaves.

Clay soil is a beast to dig but it contains lots of nutrients, especially the black clay. When I gardened in black gumbo, I used a lot of soil conditioner/landscaper mix on top of the clay and kept it mulched and fairly moist. To my surprise, a few years later, I could easily dig with a hand trowel in the beds that I improved like that. Are you trying to overcome the PH with greensand? Another useful tip when excavating for a shrub or tree is to spray the area with soapy water a day before digging. I'm not a chemist, so cannot state why this works but something happens to release the clay particles and it's easier to dig and doesn't stick to the shovel.

We have alkaline clay here in north central Texas, too; even our water is alkaline, & our summers are hot & dry. I've had great results with sheet composting. I just spread out OPBL (other people's bagged leaves) & put vegetable peels etc under the layer of leaves. When neighbors bring me their lawn clippings, I spread them out & add leaves or wood chips on top. When I get the opportunity, I add pine needles, sawdust from a woodworking shop, cedar shavings, etc. Over time, the materials seep in & improve the clay, & they've even raised the level of the ground, which improved drainage. When you start with slow-draining alkaline clay, you just can't go wrong; anything is an improvement! I also do not plant anything that *has* to have a more acid environment. no pine trees or azaleas or trilliums!