Yes, something other than a CO2 change can kick off the initial warming but then the oceans start outgassing more CO2 and H2O. Since the CO2 is the non-condensing of the two, this will continue on in a limited positive feedback fashion until it reaches the S-B negative feedback limit for CO2 sensitivity. It can also reverse course at any time because if the CO2 can be liberated by outgassing of the oceans, it can also re-absorb upon cooling.
The crucial distinction that we have right now (and one that hasn’t occurred historically) is that anthropogenic CO2 is pure excess concentration that will only sequester back in the environment at a geologically slow pace.
In other words, the thought is that this process has started in motion and can’t easily reverse itself. By analogy, the climate has always had the sensitivity to accelerate in either direction but now we have lost the braking power to shift into reverse. That’s the way I understand it, and YMMV.