Trees readily plant themselves in areas they are well adapted to. Areas that don’t quickly turn into forests are generally poorly suited to growing trees.
Further, not all trees are equivalent with marginal areas generally resulting in small and often very slowly growing trees.
I am very skeptical if that's true – see formerly forested areas like Iceland, Brazil, etc. It's not easy for a single sapling to grow into a tree, even if the conditions are right. That's why companies like Afforestt exist [0].
Edit, back to your point. Forest fires and other natural events regularly kill off large chunks of trees. They need to be able to grow in those areas in order to survive.
It'll take a while to get the rest forested if I wait, though squirrels and birds moving seeds could help. Part of it is that trees can make the area better for other trees; the conifers stop grass from growing under them, which is good since the grass can be as tall as I am and block out light.
For some value of "quick", you are correct. Fertile land will indeed eventually develop into some kind of forest absent factors that inhibit such development (such as grazing cattle). I do not agree with you about this process being quick, as it can easily take more than one human lifetime for this to happen.
On top of that grasslands can store significant carbon in root systems which is then slowly released as the grasses are killed of by tree growth.
We're at a point where passively going with "let nature take its course" is not a prudent action. We need to nudge things where and when we can. There's multiple documented instances of forests being replanted, so this is not idle speculation.