LOTS of experience in this arena; I was (up until fairly recently) the corporate trainer for an organic
lawn care company.
The very best innoculant for your soil is local
compost. Simply putting down innoculums , while they are likely needed is ineffective, as you likely lack sufficient humus in your soil (your mention of chemical fertilizers is a hint, as these do nothing to replace the humus consumed by plants). I rarely saw soils with the minimal 5% we liked; chemically treated lawns are usually in the 2-3% range...
This is important because ALL soil biology lives in humus. Without this important factor, your store bought innoculum are little paratroopers being dropped into hostile enemy territory. They will be able to scrounge for a little while, but without necessary supplies, and surrounded by hostile forces, they are likely doomed...
ops:
A topdressing of compost, on the other hand, supplies a multitudinous variety of organisms (Paul Stamets says the average scoop of good soil contains a variety of species that rivals the entire flora and fauna count of North America, and compost is no different), all
native to your area, many of which will directly support your grass, and many that don't will provide direct nutrition through nitrogen amplification (smilingly dubbed the
poop loop by Dr. Ingham) and weak acid forces, etching mineral sourcing from the CEC. AND we have seeded this little ecosystem in the ready made housing our humus provides... all contingencies accounted for...
While your store bought innoculants are not a BAD idea, they are a much better idea once you have supplied habitat and food, and compost is a good way to provide those. Find a reputable supplier you can look in the eye and shake hands with (both mine are local farmers that I have vetted their operation, looked over their product, and even glanced at it under a microscope). Local is alway better, but many bacteria and fungi are circumboreal, so even not so local is better than none. Bagged prodcuts should be avoided for the most part; these are aerobic organisms for the most part, and YOU try breathing in a plastic bag for just a minute and see how you like it
(There are exceptions; some better companies are microprickng bags to increase breathability; Organic Mechanic, Coast of Maine and North Country Organics all fit this bill).
You can get some companies to blow a nice even layer for you, but that is pretty expensive. Simply spreading compost about a quarter to half inch deep, working it in past the crowns of the grass with the back side of a lawn rake, and watering in deeply will get you innoculant (thousands rather than two or three species), humus AND nutrients, all in one package, Not bad, huh?
HG