I'd argue that the two are not mutually exclusive, you can be both. So much of what the SM does can be seen as prescribed, task-driven, automatic, etc... In my opinion the best stage managers are the ones that can accomplish all of this and still be seen as a person and not as a robot. That is where part of the art of stage management comes in to play: managing the myriad personalities and situations an SM sees daily requires you to be human. I no longer SM, however I regularly use personal bits about my life to build relationships with both clients and colleagues. I'm quite selective, and share based on where I know their interests, problems, and likes/dislikes fall. Adding a personal touch to daily situations an SM encounters is an invaluable way to build relationships that ultimately make your job easier.
Now, I don't think that it's inappropriate to talk about your workload, however if you do bring it up be deliberate in the way you broach the subject. If you're framing your discussion from the point that you feel the production would benefit if x, y, z duties were handled by someone who may have a bit of spare capacity that is wildly different than saying "Holy crap I am so busy it's driving me insane!!!".
If you need to vent, and we all do, I'd keep that in your personal circles and out of the production; however, if you're legitimately stretched too thin it's both your personal and professional responsibility to bring that up. As long as you can frame the discussion as a net gain for the rest of the production I think you're in good shape.