This simply opens a "hole" for new trees to grow up around them. It's not that any one tree holds carbon forever, it's that an acre of forest in dynamic equilibrium holds a lot more stored carbon.
When people suggest planting trees, the process isn't one-and-done. It's actually re-afforestation, where the land is permanently forested. Otherwise yes, if you cut down all the trees and then also prevent anything from growing back in its place, then of course you're undoing future benefits.
There is also 'mild' re-afforestation, where you simply increase the equilibrium amount of stored biomass in a landscape. You can plant 20-40 trees per acre in agricultural landscapes (this actually improves crop yield if you choose the right species, search for "farmer's trees"), or re-green suburbs that have lost their vegetation cover over time.
These projects are less stark than Atacama-desert-to-Amazon-rainforest, so many folks will neglect the possibility entirely and therefore conclude that "the only solution is to green deserts." :-\
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It's also probably related to cultural biases that view deserts as useless, worthless, and most important uninhabitable spaces. However if you actually try to green a desert, you'll suddenly find lots of terribly inconvenient "desert people" actually do live there, and oppose you destroying their home and displacing them. See the real reason behind building "The Line" in Saudi Arabia.
The only solution is to green deserts, which you need nuclear power for water desalination to do.