There actually is a much easier way with enhanced weathering. Igneous rocks naturally carbonate as they weather, and pull CO2 out of the atmosphere to make carbonates. This is why when you have a mountain building event it causes global cooling. So what you need to do is expose more igneous rock surface area to the atmosphere by grinding it up and spreading it out. This also costs energy but not nearly as much as carbon capture, and it's also slower. But we know it works, and there are several pilot studies trying it.
The problem is capitalism. There's no room for a zero-profit process in the economic system that everyone accepts as necessary. It has to somehow enrich the investor class.