My experience has been different in this. The least expensive first move and most bang for the buck will come from humates to make sure your soil has a broad range of the necessary trace minerals (sorry, Dave, no offense but my experience tells me this is necessary) and molasses to support and quickly increase your microbe population. Both can be applied in liquid form. The chemical fertilizers that have probably been used on that soil for years has likely locked up or stripped many trace minerals and compromised the microbe population. I could be wrong, but it's inexpensive enough to make sure that it's worth the added effort.
While the grains will add organic material and compounds that will help loosen the soil and feed the microbes, especially the beneficial fungi, it is not in itself a complete nutrition source. You'll need some balanced fertilizer and you can get it in liquid form (best choice for large areas) in an affordable form from several sources. If it were me doing this again (I helped in the remediation of my kids' school field for soccer a few years ago), I'd do liquid as you suggest and the first call I made would be to either Brad Watson @ Watson Ranch, (Brad is a frequent contributor here on the forum) Medina or Garden-Ville though there are others with good products out there. There are distributors who do business in your area, my personal favorite being Natural First in NW Fort Worth, and a couple of others.
Physical aeration prior to application of these materials will enhance the process, but it's not mandatory for them to work. Considering you're working in that clay soil, aeration should be relatively easy and I'd do it if it were me. The kids playing on the field will pound it down, and you'll need a healthy soil to counterbalance that activity.
Good luck, God bless and thank you for taking the time and making the effort to do this for the kids. They'll appreciate it.