The Bible does not prohibit doctors, nurses and other life-saving emergency personnel from working on the Sabbath. Even Jesus confronted the Pharisees that Jewish law did not prohibit, things like feeding the poor, taking a sick child to the doctor as a violation of the Sabbath. When Jesus healed the woman who was bent over, and the man who suffered from a withered hand, he did these things on the Sabbath, and demonstrated the hypocrisy of his enemies to criticized Him for it.
In ancient Jewish law and even today, Sabbath observance is trumped by saving or preserving human life. In fact, you are more in violation of the sabbath by not helping a person in need on the Sabbath. Taking care of sick person, rescuing someone from a burning building are not violations of the Sabbath by any stretch of the imagination, so one cannot really use emergency personnel as an argument against the sabbath.
Absolutely spot on - well said and bless you.