This is a well understood process and has been under intense investigation for decades. What appears to be new here is some means to reduce the cost of this "enhanced weathering" that can allow for improved quality of croplands while producing a means to lock up CO2 into the soil. There's no real downside here, other than the cost which is the point of this scientific study.
If you read the second link in the Summary, it doesn't quite share your lack of downsides. And it is optimistic. Carbonate overload in soils has long been considered a problem for growing crops, and even the huge amounts of powdered silicate rocks will probably add silicosis to the problems of wildlife and humans.
In the end, the basic chemical reactions are pretty well understood indeed. But the large scale effects of doing this stuff on a global basis are not. So we have to take parts we do know about and try to extrapolate, along with the possible unintended consequences.
A lot of information can be gleaned from acidic mine drainage, which can be treated with Calcium Hydroxide, Calcium Oxide, Sodium Hydroxide, Sodium Carbonate or even Ammonia. It works, but is expensive (we do a fair amount of remediation north of where I live in PA)
We're going to get in the weeds a bit here, so hang on...
One of the major differences between Sodium carbonate and bicarbonate is pH. The good news there is that bicarbonate has a lower pH. 8.2 vs 11.6 for the single version, carbonate (yikes!) Bicarbonate is also very soluble in water.
So if acidic rain (usually sulfuric acid) reacts with Sodium Bicarbonate, it releases Sodium Sulfate, Water and releases CO2 gas. There are still places where acid rain is an issue.
But even if it is just the carbonic acid naturally in rainwater, the reaction results are Sodium carbonate (not good) water, and release of CO2. So while there might be a sequestering effect, the normal acidic reaction tells us that a fair amount of the Carbon will be re-released, and the very high pH sodium carbonate will be produced. This is bad for the field, and then there are the runoffs into the creeks and rivers nearby. If in a creek contaminated with acid drainage, it will actually be good, if in a normal creek or waterway, perhaps not so good.
For other carbonates, the reactions are similar, in the end, CO2 is released.