The opposition between language as a system with grammar and syntax, and language conceived as a system of signs that convey meaning gets back to the need to decide on a philosophy of language.
I suspect linguistics as a field decided early on to skip all that stuff and get straight to language as is. Cut the gordian knot with the brute force empiricism. After several decades of empirical work the field has proliferated into subfields with no overview ... because empirical study only goes so far without such an overview to organize results into a coherent picture.
Kenneally and others are trying to provide that overview by linking linguistics up with biology. My opinion is that's fun, but these linguistic types have no idea what a wildly diversified, differentiated field they want to join in the biological sciences.
If you make a list of the sub fields of linguistics and then make a list of the sub fields in biology ... you get chaos.
You need to go back to basics and sketch out a rough theory of perception and communication that leads to a theory of mind that is sufficiently ordered and flexible to get into a discussion about language and biology.
Language is a biological phenomenon. The central issue is locating it within a global picture of how it fits within animal behavior. So skipping a bunch of steps what are animal brains (seat of behavior), what are they for, and how do they work? To answer those questions you need a theoretical model, and a philosophy of mind---that is designed in advance to comprehend language and other symbolic activities, including such specialities as mathematics.
CG