To me, that is a separate issue altogether. I can understand the fear/uncertainty in feeling like your job is insecure, but one sure way to make your job insecure is to refuse to help out when needed or to be dramatic about it. She has certainly framed this situation as that she doesn't think she should have to answer the phone and that she is insulted that they would ask her to do so. If what you state above is the underlying issue then the answer is simple - start an aggressive job hunt.
Asking someone to sit at the receptionist desk (and actually in this case multiple someones, not just one person) for 1/2 hour a week or what have you, is not the same as demoting someone to BE the receptionist. If you're job hunting you never even have to mention it, I don't see how it would damage your career. The perception that you don't want to be a 'team player' (forgive the lame cliche) will certainly damage your career, however.
ETA - Another reason that I would absolutely pitch in and answer phones is that customers/clients are the primary reason any of us HAVE jobs for the most part, so having someone to answer the phone is a critical part of daily operations. Sending people to voicemail is not an acceptable option in my industry and it seems not in violets' either. Personally, I'd welcome the break in the day to be able to just answer phones and chat up some visitors, I might learn something new or gain a different perspective on how we're perceived as a company and be able to improve some things.