Probably lack of water, but it's up to you if you want to address it.
Takes time, a good start for dandylions is a sprayable weed killer in the pump bottle, you buy it undiluted and add water, walk around your yard and spray those on contact. The weed killer you used does look like it's working, but the spray shit on contact really works. Overseed in the fall, and water in the spring if you want to promote grass and not weeds. It's a tough ball game in direct sun like that with dry earth.,.. and just depends on how much water (if any) you want to spend on that area of your lawn. I refuse to pay for water to water my lawn, so I have 2 rain barrels in the back yard and they do a decent enough job on the dryest parts of my lawn, which are on a slight hill in direct sun about 70% of the day. It's not thick turf, but it's green and a mix of grasses, clover, some ground coverage, and some flowery weeds (The purple ones in early spring). It's good enough for me.
FOr that spot, when you turn the soil or agitate it (you can try an aerator too, not a powered one, there's manual ones that cost like $15, they dig up plugs of earth and let water, oxygen, nitrogen, seeds, fertilizer into the lawn), and then over-seed with sunny grass mixture, you could try putting some peat moss down on it and mixed into the soil. you can buy bags of peat moss for basically nothing at the hardware store. When you plant grass, toss some peat moss onto your soil with the seed, water it, and lay some peat moss down on it it'll keep it cool and moist, protect the young grass... You just want a thin layer, not clumps, loosely hand tossed on top of it. Peat moss costs next to nothing, 1 large bag can do most good sized yards esp if you're just doing patches.
I have the opposite problem on another part of my lawn, basically no direct sun at all, and so that's pretty heavy moss. I dug up the moss this year, turned over the soil, spread lime down, over-seeded with shadey grasses (Jonathan THumb brand, reasonable priced at most hardware stores in the US), and it's actually taken this year. Still young but started growing a good amount of grass int hat area. Watered it lightly, daily, for about 3 weeks. and now I'm letting nature do it's watering from here on out. Lime increases the PH of your lawn which helps slowly reduce moss. It doesn't kill the moss, but it makes an environment that moss doesn't like to grow in and grass does.
Lime
Peat Moss
Seed
Water
Nitrogen
All cheap, DIY shit that should improve most lawns... you won't have one of those PERFECT GREEN catalog lawns, but it helps and takes almost no effort.
