Both. See the followup here: https://rootsofprogress.org/cement-redux

Yeah, you gotta dig the stuff up, heat it to high temperatures and then ship it to where it is used.

Dig it with electric mining tools, heat it with a solar furnace, and ship it with electric trucks?

Maybe, but that's not how it's done today, and it's not the sort of thing you change overnight.

Complicating matters is the fly ash component of modern concrete. Fly ash can replace 50% or more of the portland cement in concrete. Fly ash is a byproduct of burning coal. So when you replace the coal plants with solar arrays, you need to create even more portland cement than you would need to if you were burning coal (and using the subsequent fly ash in your concrete.)

I've read that kilns used to produce the clinker are often fueled by burning waste like used car tires.

Wikipedia speaks to it a bit:

https://en.wikipedia.org/wiki/Cement_kiln#Alternative_fuels

The latter; slaked lime will reabsorb the same amount of CO₂ that was released in its calcination, but portland cement absorbs much less. Moreover, the amount of CO₂ released in the calcination process itself is much larger than the amount released to produce the energy for it, even if you use fossil fuels.