Erm, that's one way, but a lot of effort. I wouldn't rake it because the thing you need to do once you've seeded bare patches is spread cut straw to keep the moisture in (like a mulch for the grass) until it can get going. Your cut grass is the mulch the seeds need. It should also protect the grass seed from birds.
I'd say get a soil test to see if it needs lime.
If your husband can *stand* it, seed it with white clover. It will fix nitrogen and help the grass grow better.
I'd get rid of the fertilizer in the shed. Temptation to your husband, just sitting there waiting for him to have a weak moment. The more fertilizer you add, the faster the grass grows and the more you need to mow. Letting the clippings decompose in the sward, IMHO, is enough fertilizer. In an ideal situation all you'd need to add is lime from time to time to keep the area slightly alkaline (grass likes that better).
And I'd be concerned if there aren't weeds. Like, what kind of chemicals were applied before you got there? Any chemical-free lawn should have dandelions and plantain, imho. Tell the dear man that you are Permies now! Dandelions and plantains are edible and medicinal! You're growing FOOD in the lawn, with no effort!
