[RigglePLC]

Compost on top of the seed is fine in my opinion. But really the main factor for success with seed is water--water every day--for 30 days. And do not forget the starter fertilizer. Add "grow-in" fertilizer at 30 and 60 days.
Be sure to use high-quality disease-resistant seed. The good seed costs more--and its worth it. Be sure it is suited for your site, climate, maintenance and irrigation situation.
Don't forget to use the new lawn seeding rate. Germination is not as good when seed is sown on top of existing grass. Seed at above the recommended rate if conditions are less than optimum.

 

[RigglePLC]

I suggest Milorganite. Use the high rate for seeding. They recommend something like one bag per 1200 sqft. (36 pound bag.)
It is a whole lot easier to apply.
Open the spreader to a large setting and go. Organic. Costs less because the labor cost is so low.
In theory, you can do 1000 sqft per minute--if you have good strength and stamina.

A second possibility is a pelletized chicken manure product.
https://www.farmshow.com/a_article.php?aid=27623

Third possibility is "Pennmulch". I has a fairly large pellet that turns to a soft organic layer after rain--or irrigation. Intended for seed.

https://www.amleo.com/penn-mulch-50lb-bag/p/215557/

I am not sure if a "Breakthru" agitator will speed up the application or not.