If you are willing to start completely over, then yes, wait until August, not September. In your area the summer heat should break earlier than September but that is what you want to watch for. Evening temps will start to fall. The heat might return, but when it starts to fall, that is the start of...well, fall.

Ignore the grubs. Any damage they caused was over and done last summer. They are dormant now and not feeding. There is a test for grubs which you can do in late June. Dig up a full square foot of the surface and count the grubs. If you see fewer than 12, then cover it back up and forget about it. Normal gardens will not be affected by a low number of grubs. Don't waste your money or time unless you see a dozen or more.

You don't have crabgrass. Crabgrass is just now starting to sprout and only in full sun and only where you get continual moisture on the soil. You might have another weed but it is not crabgrass. If you can post a picture that would help to identify it. Take it in the shade or early morning before the sun hits it. Grass pictures are always better on cloudy days.

Don't know what your lawn looked like, but you likely did not need to dethatch. It sounds like you're getting advice somewhere else (a magazine???). You also do not need to aerate. If your soil is hard when it is wet from irrigation, then you may need to soften it but aeration is not the best path for that. Sure, it is sold as that, but it does not work as advertised. A better solution is to use clear shampoo. Please search this forum for clear shampoo and you'll see plenty of description about how it's done.

The only grass that will work for you in your shade is fine fescue. It is a bunch type grass which does not spread out to form a dense carpet. Each seed makes one plant. The only way to make it dense is to put out more seed. The problem with putting out seed this time of year is the crabgrass (true crabgrass) is germinating at the same time. Fescue takes 2 weeks to germinate. Crabgrass takes 1 week. You'll put out the new seed, water it daily 3x per day, see the new seed sprout in a week, and think joyously that your new grass is coming in. It will be the crabgrass. Crabgrass IS a sod forming grass that will spread into every bare or thin spot it can find. This is the reason you want to 'densify' your fescue in the fall. The only fast way to get more fescue is to put more seed down...in the fall.

Testing your soil is probably a good idea. You seem to be anxious to do it right. It is very easy for someone to give you advice to apply lime, but only a soil test will tell you how much, when, and what kind.

Here are the basics of lawn care

2. Water deeply and infrequently. Deeply means at least an hour in every zone, all at once. Infrequently means monthly during the cool months and no more than weekly during the hottest part of summer. If your grass looks dry before the month/week is up, water longer next time. Deep watering grows deep, drought resistant roots. Infrequent watering allows the top layer of soil to dry completely which kills off many shallow rooted weeds.

3. Mulch mow at the highest setting on your mower. Most grasses are the most dense when mowed tall. Bermuda, centipede, and bent grasses are the most dense when mowed at the lowest setting on your mower. Dense grass shades out weeds and uses less water when tall. Dense grass feeds the deep roots you're developing in 1 above.

4. Fertilize regularly. I fertilize 4 times per year using organic fertilizer. Which fertilizer you use is much less important than numbers 1 and 2 above.