Bare spots might be fungus or grubs. So a fungicide and Diazinon product there might not hurt.
IMO, spread with 2-4-D/fertilizer mixture. For even application, set the spreader to 1/2 the recommended setting, then roll the lawn twice in two different directions.
Then I'd go on walkabout every few days with 2-4-D spray spot treating. 2-4-D would work well with fertilizer, as it's a synthetic plant growth hormone analogue which makes them out-pace themselves. The fertilizer only makes it worse.
I'm lazy, and unless the lawn patch is the size of a postage stamp, I'd just keep seeding, 2-4-D'ing and watering, maybe thatching a bit. Get a guy out to roll plug/aerate the turf. The whole kill-em-all. Till. Reseed, cover with hay etc. is just too much work. Can't walk on it. All soft and muddy, big rainstorms washing away the grade.
There's probably more grass there than you could grow to your satisfaction even when you remove all the weeds.