The assumptions for Corollary 6.18 are not just the ones stated in the question. They are really a whole list of sometimes rather technical assumptions on the source and target category. Still, they are satisfied in many cases of interest.
Now about stability. On top of the page 2909 of our paper (the same page of Cor. 6.18) there is a diagram with name (6-2). It is a commuting square of Quillen equivalences with the category of $n$-homogeneous functors on the bottom right. Corollary 5.25 states that the category on the bottom left, multilinear functors with values in spectra, is a stable model category. This proves that all the other model categories in Diagram (6-2) are stable as well, in particular we have Corollary 6.18.
The way this is proved follows very closely the arguments given by Goodwillie in "Calculus III: The Taylor Series", https://msp.org/gt/2003/7-2/p04.xhtml.