The point is the carbon capture plant shouldn't get any power. For sure there is no point in running direct air capture from carbon-emitting power sources. (It may, but very unlikely, be worth running it from some fossil fuel station with carbon sequestration of its own, but these in effect don't really exist outside of a handful of not particularly successful test projects. Though direct air capture is kind of in the same position at the moment, so it's all basically just experimenting with the technology at the moment.)
This might make sense if there were a single big but scarce global bucket of instantaneously available energy. But that's not how it works. The availability of energy is incredibly skewed across location and time.
Indeed. I wasn't insinuating it never made sense to power something like this today, just that it only makes sense when there is excess renewables available.
Yes but there are excess renewables available, in many places at many times. So it's already the case that it makes sense to power something like this, in many places at many times.