Are you kidding me? The police, contrary to your assertions, do not carry arms to try to eliminate "that criminal element". If you will recall we have this little detail, the presumption of innocence, that makes that a bit illegal.
Why police carry firearms is precisely why any citizen would, to protect their own life and possibly that of another. While I can legally shoot someone stealing my car (or wallet), a police officer cannot. So, in that sense, a citizen is
less limited in using deadly force than a LEO. You are correct that police are more often called to the scenes of violent crime, yet for each such call they face no different threat than the victim that they seek to assist. It is not the frequency of one's encounters with criminals that a self defense weapon is about, it is the ability to stop it, which is no different for a LEO or J. Q. Public.
Simply because you drive less miles than I do makes no difference, we are both expected to have liability insurance. Think of a self defense weapon as that insurance, you hope to never need it, yet it is far better to have it and not need it, than to need it and not have it.