a modest natural top-of-atmos-phere radiative imbalance of ~2.6 W m⁻² can account for the observed rise in atmosphericCO₂ via temperature-driven oceanic outgassing and enhanced soil respiratio
The TOA radiative imbalance does not affect CO2 fluxes, directly. It only does so indirectly, by affecting temperatures. We know from ice core data that both over glaciation cycles and over shorter climate cycles such as the RWP 🡖 DACP 🡕 MWP 🡖 LIA, 1°C of average temperature change could only affect atmospheric CO2 levels by at most 10-15 ppmv (and even that was only after long equilibration periods).
But the atmospheric CO2 level has risen by 112 ppmv just since 1958 (when precise measurements began).
Obviously we haven't had 112/15 = 7.5°C of warming since 1958 (and we also haven't had hundreds of years to equilibrate).
What's more, CO2 levels continued to rise (at an accelerating pace!) during the 1950s through 1970s, as temperatures FELL. So obviously the rise in CO2 levels wasn't caused by rising temperatures. (It was caused by the post-WWII boom in industrial production.)
What's more, we know that both the oceans and the soils (as well as the terrestrial biosphere) are currently net removers of CO2 from the air, not net producers of it.
The sole reason for the ongoing increase in CO2 level is that human CO2 emissions currently exceed the net rate of natural CO2 removals.