I agree that growing from seeds is hard. Different seeds take different times. Weather & temperature matters a lot too. You never want to seed in the hot summer or cold winter. So Fall & Spring is the best period to grow grass.
You want to take a look at the weather forecast for next 2 weeks that range between 60 to 85 that have some sunlight. Ryegrasses take at least 10 days. Bermuda grasses take at least 15 days and Bluegrass at least 30 days. And you need constant watering. Germination require perfect condition. Sometimes the germination won't happen this year, but it probably will germinate next year when the condition becomes perfect.
If you want the lawn repair itself, you need species like Bermuda, Bluegrass or St Augustine. If what you have is one of the above, you can repair the lawn like the following video.
Ryegrass or Fescue won't repair themselves.
Buying sod from local store will be a lot faster, but the grasses you got very likely will look different from what you already have.
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The reality is, there is no easy way to fix a lawn, it takes time and some effort.