For using power to suck carbon out of the air, would you need baseload power, or could you have a machine that ran whenever cheap solar electricity was available.

Removing CO2 is important, and urgent in the medium term, but it isn't urgent in the "must happen this second" sense. Intermitten removal would be fine if the removal tech could be power intermittently.

Most steps of an industrial process will work with intermittent power, albeit with substantial loss of process efficiency that will therefore require building more infrastructure for the same effect than if you were using baseload power. Some steps of an industrial process may effectively require non-stop baseload power because restarting it is prohibitively slow and costly.

It may be possible to do atmospheric CO2 removal with intermittent power, I haven't attempted any kind of chemical engineering deep dive into what that process entails, but that is likely to incur a significant loss of productivity. Given that it is already an extremely expensive proposition, we will probably need to maximize process efficiency, which means reliable continuous power.

Hydropower is actually not so nice when you take into account that it's the reason that many fish species are on the brink of extinction. That's in Europe at least but it seems that it's also a problem in the US, see https://www.vice.com/en_us/article/gvm7vq/power-plants-are-b...

It will also start to have problems as temperature further increases, requiring fully closed storage systems rather than current use of rivers and aquifers.

That changes economics a lot.