To grow a good stand of grass, or most any other plant, you want a soil pH in the 6.2 to 6.8 range because that is where most all soil nutrients are most readily available. 5.9 is a bit low so adding the right lime, calcitic or dolomitic, can help your overall program. Put on compost, or shredded leaves, or something similar, preferably not the bark because that is just not something the soil bacteria will easily digest unless you have a good, healthy soil already. Sand, becasue of the large particle size, does not cause soil compaction, although small amounts mixed with clay can cause clay to bind more tightly together. That is why few of the Ag Research Stations recommend adding sand unless you add at least 45 percent by volume (an extremely large amount of sand). Since this is to be a lawn I'd not be concerned about the Potassium level, once sufficient quantities of organic matter are in the soil that will come up, in my experience. Once your have sufficient levels of organic matter in your soil compaction will not be a problem, so you should only need to aerate this once even though it can take a few years to get sufficient levels of organic matter intothe soil. Rototilling your lawn will leave you with large, bare areas for some time. Properly amending the soil will get grass, as long as what you have is what you want, growing faster. Whether rototilling the grass you have into the soil or not is good or not so good is always the question. Some grass species are very invasive and rototilling can often simply spread the grass roots around and give you more grass, but the grass does add organic matter, so it depends on which grass type you have nad how much future work you want to do. With my Quack Grass simply rototilling creates much more future work, because of the amount of grass that will grow back, than covering the area with newspaper and a mulch material which will kill the existing grass and most of the roots, even though I know I will still get some of that grass growing in that bed in the future. However, the soil is workable enough that pulling those grass plants and roots is very easy. Often a cover crop, of one of those cover crops with allelopathic (plant growth suppression properties) properties can help.

Sterilizing the soil is a romantic but impractical notion. It's not possible to kill seeds of unwanted plants without also killing all of the important wildlife found in healthy soil. Very few plants will grow in "dead soil," and new seeds will blow in & be spread by various critters. Smothering is much better, for preserving that. Whatever method one uses or starts with, unwanted plants will constantly be showing up. Without periodic patrols to pull them out while small and easy to pull, a "weed problem" will develop. You may not think soil microbiology is interesting, but if you want great plants, you need great soil: http://permaculturenews.org/2013/09/20/soil-not-dirt-dr-elaine-ingham-talks-soil-microbiology/

I swear nobody around here has heard of core aerating. I want to move there! Far too many people here think it is a panacea and solution to hard soil. Instead of clinging to that crutch, try this. Spray shampoo at a rate of 100 cc per 100 square meters. Your English is so good I'm going to assume you're a Yank living overseas. That converts to 3 ounces per 1000 square feet. Then water 1/2 inch to 1 inch and give it 3 weeks for the magic to happen. Since you're already feeding with organics, it should work quickly. I think your soil test is going to provide the solution. I am often luke warm on soil tests unless you've done all the organic stuff and it didn't work. Something isn't right and I'm going to guess it is salt balance. It could be something worse like too much boron or copper in the soil. The soils in the upper desert of Southern California are too rich in boron to be a good place for plants. Those micro nutrients are good up to a point and then they become a poison. Waiting until May should be fine for fertilizing. Fertilizing early with organics should not have been a big problem. The way they work the microbes would have begun slowly converting the protein into plant food for the plants as needed. Do you reseed the thin areas every fall? If your climate is anything like western Washington, you might be able to seed in the spring and not get summer weeds. How warm does it get in the summer?

One thing you can do, is add about 1/4" to 1/2" of clean sand at a time. The existing turf will re-establish itself at the new height over time, and then continue on with another layer of sand. Adding "dirt" or soil can be problematic if it is of a heavier or clay type. The idea is not to add a layer of poorly draining soil on top of well drained soil. Sand of course drains well, but it will incorporate organic material over time and become good soil. Fall is THE time to do this though, trying to get turf to grow in the spring is kind of a rookie move. As soon as summer starts to wane (late August round these parts) prepare your seedbed, put some seed down and fert and water it in every day for a couple weeks. It is the only way to go for cool season turfgrasses. The problem in the spring and summer is every friggin' weed on the planet is going to compete with the young seedlings. In the fall the temps are more conducive to bluegrass germination and growth and the weed competition is far less.