Not to nit-pick, but police are really not like everyone else in that their training and qualifications are very specific to their job of policing, including handling guns.
I don't think you are nitpicking.
I do think you are operating under a misconception of police training. In most municipalities, police are only required to qualify with their firearms once or twice a year (outside of SWAT, etc). In fact, the range I am a member of is where our local police force trains. They have two training days per YEAR set aside for them.
Some, a very few of them, train weekly or monthly and I have trained with them. Most go far more rarely or the minium. In a larger city I used to live in, it was worse (my friend was an officer and I sometimes went to their firearms training facility).
The fact is, a LOT of civilians such as myself are far more trained on weapons that most of the police you see. I have a higher rate of on-target shots that the police are required to have. I regularly shoot, disassemble, clean and put back together my firearms. I know the quirks of each one inside and out.
We also have a specific officer who is known for pulling his gun often as he gets protection from the city. There is a pretty well known event where he pulled his gun on and arrested a woman for watching, from her own front porch, him arrest someone after he told her to go in her house. Cops are no better or worse than anyone else.