Answer: Based on your description it sounds like you are trying to do things right. Keep in mind that summer is just not the best time to plant a lawn -- but early September is perfect. There are many factors involved in starting a lawn. And any time there is bare soil, nature will try to cover it (with weeds) and weed seeds can be blown in. Every time you till or disturb the soil, more weed seeds are brought to the surface where they can germinate. And yes, they can be imported with top soil as well. And, top soil is not a regulated product so you have no real way of knowing what the soil is composed of.
I am concerned that the soil has turned hard; ideally your soil should have ample organic matter in it to prevent that. Correct watering should also help keep the soil from turning hard and compacted.... The basis of a healthy lawn is good soil conditions. Since you have a little time, I would suggest you work with your local county extension to run some basic soil tests and begin with preparing the soil based on the test results. You may simply need to adjust the pH or fertility, or it may require additions of compost. Since you have already fertilized you may not need to do that again. The only way to know is by testing, then you can work with them to develop a plan.
Crabgrass is an annual weed, so it may be that your perennial grasses will crowd it out. The desirable grasses may also crowd out some of the other weeds you are seeing -- if you can keep conditions ideal for the grass and have planted a grass variety suited to your local conditions and soil. Spot treatment is also an option, to try to preserve the good grass and remove the most invasive weeds while they are small and before they have gone to seed or spread.
I would not recommend aerating a newly seeded lawn. A spike style aerator is not helpful in any case. But aeration might be something to do next August/September if you use a core aerator and pull out plugs of soil, then rake them off and replace the material with compost or something like that. Instead, this year you might be able to overseed in early September with a slit seeder if the stand is really sparse. That way you will not be starting from scratch and can preserve the grass that you already have.
Again though, this is something to work out by consulting with your county extension -- they should be able to help you do some turf analysis and identify the weeds and help you plan how best to approach the problem.
Best of luck with your lawn!
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