Changed my PATH to have /usr/local/bin ahead of /usr/bin and now pip isn't found
Ok... so after a bit more digging I think I found the answer to my own question. While running pip --version doesn't work, running pip3 --version gives me pip 9.0.1 from /usr/local/lib/python3.6/site-packages (python 3.6).

What I think is happening is that pip is located in the same location as python2 that comes with the Mac OS, while pip3 sits in the homebrew location.