Second largest industrial producer of CO2. Shipping the sand long distance also makes it worse.

And it's also impractical. Our issue isn't the absolute availability of sand, it's the easy availability near use sites.

It’s not easily available near all use sites. Australia exports a lot of sand to the Middle East and South-East Asia, for example.

And even if the energy input were totally renewable (or nuclear), the CO2 production would still be pretty large, since one is decomposing limestone into calcium oxide and CO2.

It is too bad more isn't done with lime mortars and plasters. Hydrated lime (CaO + H20 => Ca(OH)2) reacts with CO2 to form CaCO3 + H20. And it is fun to work with ...

That can, at best, absorb the CO2 that was emitted when the CaO was made, if the CaO was made from limestone. To get CO2-free CaO would involve decomposing calcium silicate, perhaps by acid dissolution followed by the energetically expensive step of separating calcium chloride into lime and hydrochloric acid.

Yes... but another advantage here of lime over Portland cement is that the cooking of the limestone doesn’t require as much heat as the cooking of Portland cement. It won’t be carbon neutral, but it is a big improvement.

Isn't that the same limestone that can capture carbon if put into the oceans? or is it olivine?

Olivine is usually discussed. However, I think limestone would enable some CO2 uptake, by formation of bicarbonate ions. The calcium ions would have to stay in solution, though.