I'm pretty sure the answer is that your pH is too high (that depends on who you ask) and especially that your nitrogen is too low (I think most will agree on this). If you have high potassium and low nitrogen that will probably inhibit germination and growth.
People generally lower the pH with sulfur, but that can take a lot of time and it isn't always the best solution according to some. I prefer mixing/tilling peat moss in with the soil to lower the pH, since it adds organic matter, too, and is bulky. I'm not into lawns much, but I'm thinking ammonium sulfate, urea and such might be good for the nitrogen as might different nitrate fertilizers (but calcium nitrate would be a bad idea since your calcium seems to be high from the lime, and calcium nitrate is said to kill soil microbes).
You might also try growing inoculated clover in with your grass (if you don't mind extra bees in your grass), as it should help to increase nitrogen. Or, you could do only clover.
Grass is supposed to love nitrogen.
I must admit that I don't know for sure that these measures will increase germination rates, although I believe the organic matter from the peat moss should help considerably, especially if the soil drains too fast. Composted manure may help too. People or sites might tell you not to use a high nitrogen fertilizer, but because your other soil nutrients are high, you need to balance them with extra nitrogen.
In my area, people usually put straw or hay over new grass seed. It helps to keep the soil moist and probably eventually supplies natural nutrients. Straw is high in nitrogen. It is said to protect the seeds from being blown away or eaten by birds. I don't know if you did that, but it might be a good idea.