If you are not opposed to using Weed-b-Gon Chickweed, Clover, and Oxalis Killer, then use that. Be sure to read the label concerning the temperature to apply. It should have been applied in mid April. If you want to wait, you could apply in September, but with the dogs you might not want to wait.

Pulling long runners is very effective for the living plants. The problem is that clover drops tons of seed, so it seems like you can never get it all. Even with spraying, it only kills the living plants. You have to spray again as soon as you see new clover and before it goes to seed. By that time it would be spot spraying.

Dethatching will do nothing against clover. There are already seeds in the ground. What will help is mowing the grass at the mower's highest setting (you don't have bermuda do you?), and watering deeply and infrequently. The tall grass helps shade the soil where the seeds are, and they need sunlight to germinate. Withholding water does not allow the seed to stay moist continuously long enough to germinate. Of course Mother Nature will come along and toss you a 3 or 4 day rain event which will germinate all the seed She can.

I'm sure that the soil was very compacted, which I understand is great for clover

You have that backwards. Clover, with a plethora of vigorous roots, is very good for loosening compacted soil.

Unless you are getting into the lawn business, you don't need to buy a dethatcher. You can rent them at Home Depot or other rental stores. Same for core aerators, which I don't believe do a dadgummed thing, but others disagree.