The State Department is highly competitive and how you score on the Foreign Service exam tracks you. Is the Post Office not a Union gig? Police Officers start at a very low wage in NYC like 25k per year or so. So I have trouble imagining that a mail carrier earns 'high' wage. One of the best wages one can get in New York of that nature is in Sanitation. I'd imagine it'd be kind of depressing to do such a job after working so hard on an engineering degree.
When you talk like this I cannot imagine that you have much daily contact with these sorts of professions. The idea that they are not competitive is divorced from reality. Interviews are usually done by Cattle Call and you'll be among about 200 people applying for the same 6 jobs. The same is true if you want to work for UPS, or for NYC Transit.