If there are no herbivores or regular fires, forests will pop up, but not all forest systems are equal.
The boreal forest, as much as I appreciate it, is a coniferous desert. If you compare it to temperate hardwood/boreal forest transitional zones, the amount of biodiversity in the former, both faunal as well as floral, pales to stark whiteness compared with the life explosion of the latter.
As an aside, the comparison of the boreal biome with that of deserts shouldn't be surprising; needle-shaped leaves and waxy coatings are, after all, evolutionary adaptations to periods of intense aridity.
Also, forests without a successional pattern become artificially senescent, or older than they're biologically supposed to get. This is different from long-lived forest systems, whose ancient trees might continue to actually grow, accumulating biomass and trapping carbon, for centuries. Carbon sequestration drops off with the drop-off in biomass accumulation, as instead of growing trunk and branch, growth energy is relegated to seasonal growth related to reproduction.
And many of the food plants, in the boreal forests in my neck of the woods, anyways, are disturbance-related. If you want blackberries, raspberries, and blueberries, often all you need to do is visit a clear-cut or burned-over area in the following seasons.
I think what you see a lot of in these disturbance-free green deserts is the slow build of fuel reserves, just waiting for fire events to finally allow succession, with insect infestation to dry things out and push them along.
The point about the function of deserts in rain cycles is valid, but the point behind the greening the desert movement, as I see it, is to reverse the trend of desertification. This has been an issue since the first wooden plows, stone axes, and domesticated and feral goats and cattle. It might be well and good that we need some amount of desert. We don't need them to increase in size. We don't need the desert to swallow farmland so that more land is cultivated, for the cycle to continue, leaving us with dustbowls and famine.
The reason, as I see it, that prairie and grassland is more resilient is because the growth rate of individuals is so much quicker. A individual grass plants will attain maturity and go to seed before a tree seedling is a year old. In addition, the growth of the organism is stimulated by grazing, whereas browsers kill young, and even older but immature trees, by their browsing.
So I don't think we should be down on grasslands, nor should we only tolerate them until we can get them to "succeed" (I hate that word in this context because if you follow the logic of the appellation, closed canopy old-growth forest is the apex of that "succession," or the pinnacle of success, if you will, but some of those systems could be better from a biodiversity and carbon sequestration standpoint if they didn't "succeed") to a closed-canopy mature forest.
Honestly, the only argument, aside from some shared sentimentality about enormous ancient trees, that makes sense to me with regards to not gradually replacing most of the oldest trees with young ones that have more carbon sequestration potential, is the fungal argument. Ancient forests with long-undisturbed soil layers borne of centuries of leaf and limb litter house giant, ancient fungal colonies (are they described as colonies or individual, massive organisms, I forget), and these, I think, need to be nurtured to foster health in their surrounding regions.
So I say don't worry about preserving the deserts. It would be good to be able to forecast what effect reversing desertification will have, and what effect increased desertification has had on weather patterns over the last few millenia, but I think it's most important to lock up carbon in ecosystems and in soil for now, and create resilient, food-and-soil-producing and anthropogenic change-buffering systems where there is only way too much desert now.