My reading seems to indicate that I should tear out all the plants, dig over the soil with a spade and fork, wait a bit and pull out any weeds that grow, then improve the soil with compost, manure??? then seed. Some sources say to cover the ground with plastic for a period to kill everything but I don't want to wait that long.

Does this approach sound right? Any advice would be very much appreciated. Will try to attach a photo.

There are many approaches. This one will work. The one your reading suggests will result in a very bumpy lawn after 3 years. It will also be hard to establish because of the organic stuff going on at the wrong time. If you want fast results, find some RoundUp (glyphosphate) and spray everything you want to be in grass. Wait a week and start looking for stuff that did not die. Spot spray anything green and wait another week. Then rent a power rake to chew up everything. The power rake is any machine with vertically spinning blades. You can adjust them so that they just touch the surface of the soil. That tool will shred everything. Blow or rake all that off. Then seed your seed and roll it down with a rented roller. The roller's job is to ensure good seed-to-soil contact. Finally keep the seed moist by watering 3x per day for 3 weeks. Use whatever seed for the grass you see the Belch people using near you. Don't try to get fancy.

Seeding in the spring is a bad idea but that is what you want to do. By summer the new roots will not be tough enough to stand the heat. You will also be germinating crabgrass seed, which only germinates in the spring. Magazines never mention that either.

When your new grass gets up to 5 inches high, then mow it back to 4 inches (mower's highest setting). After all the seed is up, back way off on the watering. When it is cool you can water once a month. When it is hot, up the frequency to every 7 days. If you plant a fescue blend of grasses, prepare to over seed in the fall to thicken it where it died in the heat or from the crabgrass. Next season it will look great.

After you are mowing real grass, then you can deal with compost if you want. Fresh (uncomposted) manure is never to be used in a garden. It just stinks. Use only composted manure. It smells very fresh. Still, compost is the most expensive thing you can put in the garden. Organic fertilizer, in the form of alfalfa pellets (rabbit food) is about 1/10 the cost of compost. Unless you have reason to think your soil is all poisoned, then skip the compost and go straight to the rabbit food. Apply at 20 pounds per 1,000 square feet as often as you want. Minimum of 3x per year, but if you want fast results, do it monthly starting now (after the grass is up).