I am an environmental geologist, and while I’m not going to debunk or refute the paper or author (someone more up on their game than me can), I will say that the lack of historic data was always a variable that could be reliably solved for eventually. Our fossil evidence and understanding of global tectonics was already allowing it to be unraveled back when I was in college 20 years ago.
So from a modeling standpoint, if you can repeatedly replicate what you know conditions were like in the non-ice/warm periods, you can reliably infer what the CO2 (or just overall greenhouse gas mixtures) had to have been (I won’t get into why we know it was like that, paleontologists will talk your ear off about it any day)! From there you can develop models with very robust and accurate inputs to predict how long it will take to reach those levels at current pace. Every year the trend line gets more and more granular as well because we have so much data.
Idk if/how that impacts this particular study, but it should give OP some background and trust in the modeling that’s based on data we don’t/didn’t have.
That right there is a bread!