Now this is the type of thing that is imporant to bring up. It is a huge weakness of the models as is the premise that past warming is caused by CO2 - as opposed to the reality that warming proceeded increases in CO2 levels.
The CO2=catastrophe advocates say that solar warming triggers CO2 feedback and more warming from that. For example the relatively small warming from the Milankovitch cycle plus the fact that the NH gets more sun creates the interglacials, like the present one. The CO2 in their theory is a multiplier taking 100's or 1000's of years, but basically amplifying the warming in a full feedback loop. They claim that the current manmade CO2 is a short circuit.
There are several problems with that theory. First M cycles have coincided with more drastic solar magnetic changes. Second, M cycles and alleged CO2 feedback reaches a certain high temperature and no further (a one-off event 200M years ago does not change things). Third, the feedback depends on the climate sensitivity to CO2 which depends on the solar magnetic and other factors. Those are not in the models, not in the ice age forcing studies, and an area of research looked on with disfavor.
So the modelers do agree with your statement above that warming precedes the CO2 rise, but then they add a continuous loop of warming and CO2 rise for centuries to get their result.