My procedure for this changes as there are hours in a day.
The way I like to set it up is that the maintenace crew does the power raking or hand raking, what ever is necessary. Then I follow up with couple of the men with 2 aerators. I seed, they aerate. Top it off with peat moss.
Now I will have the seed down by the time they have aerated all of it. Some will actualy be done after a strip or two is aerated. Makes no difference. Either the seed goes into the holes after the aerating or the watering moves seed into them.
That does not mean that all the maintenance crew rakes out is seeded that day.
If I have a project to do then the aerating may wait a week or two. All that matters to me is the seed is put down early enough to develope a root system by winter.
I am going to start next week. And I don't have to do all in one week. We split into two weeks sometimes 3