Just to let you know, I'm much better at cons with a topic like this.
Let's assume they do this without engineering our orbit to prevent Milankovitch cycles
The immediate question is where the water to do this is coming from, and the less obvious follow up is to ask what they did with the salt.
To paraphrase the video (it's short) water for irrigation naturally contains more salt than rain. The longer an area uses irrigation, the more salt builds up in the soil, so trying to create grasslands through irrigation will literally salt the earth. This in turn will prevent most plants from growing.
To be fair, the video points out that this problem is mostly associated with food crops like corn and wheat, due to their shallow root structures. And while it's true that some native grassland plants can have root systems just over 4.5 meters (15 feet) deep, it probably still won't be deep enough to reach the water table in an arid desert.
So without removing the salt, the con is that they would be salting the land, making any benefit extremely short-term for all of that effort.
But, what if they did remove the excess salt before using it for irrigation? We do have desalination plants already.
Well, don't believe anyone who tells you brine from desalination plants can safely be returned to the ocean.
So the other con is going to be that you're going to have a lot of toxically salty brine on your hands, and even if you turn it into baking soda as in that article, you're still going to have massive amounts of calcium chloride - which is an irritant and desiccant - left over.
There's no such thing as zero-waste industrialization.