Don't lime based mortar still require the O2 to be driven off through heating to very high temperatures? Isn't that the same reason portland cement production creates high levels of CO2? Or is the CO2 produced during the curing stage of cement?
Chemically they are net-neutral only if they are non-hydraulic lime, but hydraulic mortars do continue to absorb some CO2 over time.
Separately, they are nearly always heated with fossil fuels, but, unlike steel, it's not chemically helpful to do so.