Seems like you're getting mixed messages.

You should never rototill in preparation for a lawn. Using the landscape tractor and box blade for grading, like your contractor likely did, is the way to prep for all landscaping. Rototillers buck, jump, and dig in leaving an uneven subsurface. The fluffy soil left above that surface is invariably leveled to perfection. If you picture that in your mind the rolling subsurface has a fluffy layer of different depths. When the fluffy soil finally settles completely (3 years), it will settle to the profile shape of the underlying subsurface. That leaves your surface bumpy and leads to unpleasant mowing experience and sometimes scalped grass. The tractor with the box blade works only at the very surface. Even it they lower the digger bar the tractor will not buck, jump, or dig in. It leaves a much nicer bed for planting.

You do not need more topsoil. When it was graded it was graded with a purpose in mind. That purpose is to drain water away from all the buildings and out into a gutter somewhere. As soon as you bring in topsoil, you change that drainage. Some people believe that just bringing in 1/4 inch of topsoil every year is necessary to reinvigorate their soil. I have pictures of lawns with a full 10 inches of topsoil accumulated over the decades of bringing in just 1/4 inch. The soil you have was left pretty perfect by the landscape grader. That's his job.

I'm not sure you can wait until August to do this job but it would give you a better result. Seeding now is likely to give you a lot of crabgrass and little real grass by July. Late spring is second only to summer for bad times to seed cool season grass. Late August is better because it gives the new grass about 9 months to get ready for summer heat. Fall seeded grass also does not have to compete with crabgrass seed for moisture and nutrients.

All you need to do is seed and roll the seed down to ensure it makes good contact with the soil. If it is in your blood to apply something on top of the seed, then use a compost. Apply compost at a rate of no more than 1 cubic yard per 1,000 square feet.