I feel your pain. Just relax until mid to late August, then you can start to rectify things. For now, just let what you have (including the weeds) grow. I am assuming your turf grass is a cool-season variety like fescue (I am not familiar with Missouri). Cut it high (3.5" or so). Since you have a huge area, irrigation doesn't look like a feasible option, so allow the grass to go dormant this summer in the heat (if you get hot, dry summers). It will come back just fine in the fall.

Cutting short in summer is a sure way to give weeds the light they need to thrive while simultaneously preventing turf grass from developing strong roots and surviving low-water conditions. Also, you never want to water turf grass daily. It thrives with long but infrequent waterings.

Clay soil is a pain in the behind. Aeration is a must -- in the fall.

Soil test should be priority #1. If you have acidic soil as clay often is, you can get some lime down now since it takes awhile to affect the soil pH.

Ignore the weeds. Let them have their party for the next two months. It is too late to stop them until fall. The key to eliminating weeds is having strong lawn grass.

As for light, you may have to consider shade-tolerant seed mixes for areas that do not get much light (chewings, creeping red fescue). If the area gets little to no direct sunlight, turf grass may not be an option.

In late August, implement a coherent lawn reseeding plan that includes aeration, dethatching, adding organic matter, fertilizing, planting seed and keeping that seed moist until germination.

I understand that having 2+ acres makes some of this downright impossible. In that case, pick the area you care about most (usually the area near the house where you spend most of your time) and work on it.

I agree for the most part. I would spend this time doing what you must to identify the weed. Some weeds are really easy but others need to be worked on longer. When you take a picture, get as close as a foot away from the grass and let the camera focus. You might put a coin in the picture for reference.

If you are not going to water, then your grass should be allowed to grow long and tall in the heat of summer.

I'm always curious: why do you not irrigate or fertilize?

Photos

I have plenty of the lawn that is looking like this ..
http://sportingtimes.net/crap/garden/1.jpg

Some pretty compacted, cracked and hard ground
http://sportingtimes.net/crap/garden/3.jpg

Despite the patchy nature of things I have pretty good coverage considering the trees but I feel like I am losing that battle.
http://sportingtimes.net/crap/garden/4.jpg

The weed ... that is a 5c coin.
http://sportingtimes.net/crap/garden/5.jpg

... and as you can see its spreading quicker than Captain Trips
http://sportingtimes.net/crap/garden/6.jpg

Why do I not irrigate or fertilize ... well we just moved into the place last July, looked much the same as it does not minus weeds. Summers get pretty hot in these parts but the trees actually provide an amount of shade and so as much as it can hinder growth it also protects things a bit. It simply hasn't been necessary - I don't have a lot of browning. Throw in the size of the yard which is pretty massive and it feels like a waste of resources for not much upside.

Fertilizing simply has been a case of not dealing last fall as we were swamped just completing the move in and having a small child. I want to change this in the fall. I thought about it in the spring but didn't want to encourage the weed growth I have so unsuccessfully avoided.

In terms of adding organic matter to the lawn in the fall ... 21 trees provided me with two absolutely massive compost bins full of what I hope will be useable leaf mold. Acceptable for a lawn? If so I will have a never ending supply.

I am not sure what that weed is, but my best guesses are in order of confidence: 1) wild violet; 2) creeping charlie; 3) wild ginger. I think it looks most like wild violet but if I had to bet on it, I wouldn't bet much.

Wild violet and creeping charlie thrive in shade and compacted soil. They are both fast-spreading and difficult to control (as you mentioned) so you should probably start trying to eradicate it now.

Hopefully, someone else can confirm one of my guesses or provide a better answer.

Excellent pictures!

If you have not seen the purple flowers, then it is creeping charlie. In either case it is a pain to eradicate. The leaves look more like violet, though.

Since you want an organic solution, the bad news is there is no solution besides digging it out. If you can tolerate a minimal impact solution, you might try this: lift a piece of the weed up and drop it into a baby food jar of Round-Up. Let it sit there for a day or two and see what happens to the particular stem you have placed in the jar and to the surrounding weed in the area. This approach works for another terrible weed called bindweed. With bindweed, the roots are interconnected so killing one plant will often take out huge areas of the weed. You should be so lucky! If you are not so lucky, then repeated spraying with Round-Up will kill your entire lawn and, possibly, the weed.

If that does not work, you will have to dig it out. Start this asap because as you can see, it spreads fast. You have to get all the roots out to about 3 inches deep. It is a lot of work.

Since you reported mowing low often, I assume any flowers would be chopped off before blooming. I do think it is wild violet and I would heed the above advice on eradication.

A question you must face is whether you get enough sun under that tree canopy to grow dense fescue.