Actually that wouldn't be odd at all. If you receive a direct invitation from a CP to give me a call, that seems like an opportunity that anyone would jump on. There is a huge difference between receiving an invitation and finding out the number and cold calling. If cold calling chief pilots was a totally acceptable process, I can only assume that the thousands of applicants with a resume on file would inundate the chiefs and they would do nothing but answer the phone fielding calls from applicants morning noon and night.
I directly spoke with an assistant chief in my base that I have flown with many times a couple of years ago about getting an interview for a highly qualified check airman friend who was looking to get on with SWA. He told me he had absolutely no pull as to who gets an interview, but could certainly make some things happen if and when that individual makes it to the review board. I took him at his word and believed that he had little pull as to who is getting the call for an interview. I have heard this echoed from many check airman. I guess it's possible I simply don't have the right stuff when it comes to networking.