Broadcast spreader should work fine. In some spots, I think I can see a distinct like between the good grass and thin grass. This suggests it is due to variations in the soil under the grass. Is that left field?
I think it will work out OK. Check the soil with a soil probe. Sandy?
Also I think, starter fertilizer would be a help. Faster growth and darker green. Take care to apply it evenly. Maybe east and west-- then north and south, to reduce the chances of uneven green.
Think about a higher quality seed. The best types of perennial rye are a beautiful dark green.
Of course, perennial rye does not spread--no rhizomes--it tends to be a bit clumpy if the seeds are not close together. So be sure to get the proper amount of seeds down per square inch.
You want about 25 seeds per square inch.
And consider adding Kentucky bluegrass. The rhizomes cause the sprouts to fill in tiny thin spots--resulting in a denser turf-- eventually.
And...
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And you need to seed with the proper number of seeds per square inch. Myself I like to suggest a bit more--in case not all germinate.
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