Typically, the way you would cut a piece of land is by selectively cutting out the species and grades of lumber that are actually worth money. So one winter you go in and cut the big, high-quality pine, or spruce and fir for saw logs. Another season you might cut out the high-quality veneer-grade hardwood trees. Of course, when you do this, you would also cut a bunch of the junk that is never going to grow into profitable timber, and sell that as pulpwood for making paper, or low-quality logs for making pallets, or sell it as firewood. If you selectively cut like this, it's not particularly difficult to rotate through a series of stands and continue to harvest them indefinitely, and you maintain a variable-age forest that sustains the local wildlife, to boot.
At least on the East coast, it's not really necessary to replant cut sections, even if you do go in and clear cut them, since there is so much residual seed on the ground already, and many of the species will even sprout again from stump. You go through a short cycle of black/raspberries and other scrub growth to fix the soil, then the young softwood/hardwood whips come up through, and within ten years you've got a stand of wood that's ready to be thinned out again, to clean out the junk and let the good wood grow up into profitable timber again.
On the West coast, as I understand it, replanting is more important, since many of the species won't grow from seed UNLESS there has been a forest fire.
The United States is also more forested now than it has been in a hundred years, since so much land formerly cleared for agriculture has grown back as forest. If you could look at Google Earth imagery of New England from 1900 and today, the difference would be astounding.
Second growth is only a step in the ecological process of producing a mature forest. Most of the east coast's regrowth is not representative of the original species, though it is often farther along in the process compared to the west coast. These places take multiple centuries to come back to their former pristine state.
Why does it take so long? Plant species are sensitive to available light and moisture. Canopies regulate this on the ground, but also rely on established plants for their development. Immature, modified or non-existent canopies cannot support the same life.
Selective cutting modifies the canopy. Established growth underneath does not survive, even if it is still alive when the crew goes home. When the canopy recovers in a single century the lower plants can thrive once again.
Secondly, the question was "why don't they thin things out". Not "why don't we let them thin things out". Implying a change to normal behavior.
Thinning is more expensive and difficult than clear cutting, which is why we don't do it often.
The article says that the portion of the budget dedicated to fighting fires keeps increasing, giving less money to land management decided to prevent a lot of the fires we are fighting. Seems to me that continuing on this path will just result in ever increasing costs. We have to let it burn, on our terms.