They kind of address this in section 3.4, where the argument seems to be that if physical embodiment or some other single bottleneck prevents automation of a large fraction of all tasks, then removing this single bottleneck will enable the automation of all these tasks all at once, leading to explosive growth.

I.e. they don't predict explosive growth starting now or even in the near future, but that if automation reaches 100% at some point, there will be explosive growth in the final period leading up to that point. But somebody is probably going to misinterpret it as a statement about the transformativeness of current technology.

I think the weakest part is actually section 3.6 where they sort of admit that GDP growth does not capture many things people intuitively would consider growth (e.g. greater quantities of free stuff). They claim that this doesn't matter since they don't care about GDP per se and just use it as a proxy, but fail to address that their model predicts explosive growth even in this case, i.e. their model of GDP doesn't behave as GDP would, i.e. they picked the wrong model.