Given a population of 38,000+ the city should have a police department of about 43 officers ... if they only have 30, it's no wonder they are understaffed and overworked. 6 additional officers would be a big help. Overtime would be insufficient alone to resolve any staffing issues. At 30 officers, I would not be at all surprised if the police were letting property crimes slide and not enforcing many of the quality of life and lesser offenses that the public tends to clamor for - traffic enforcement, graffiti, noise, inebriation, drugs, etc. It's probably all they can do to handle their calls and do basic follow-up on those offenses that are a priority.
If overtime went to only a few officers, that can tend to show that the scheduling sergeant or lieutenant are helping their friends feather their beds. Or, it could be that those are the only officers able or willing to work the available shifts. In my department right now, there are three officers getting most of the overtime. Why? Because the rest of the officers are either unable to work the shifts because the OT occurs on their regularly scheduled work days already, or, they are unwilling to work the OT due to other needs such as daycare, etc. There can be more than one reason why OT goes to a select few. And with it going to 7 people, assuming it's largely patrol overtime, that probably represents almost 50% of the patrol staffing, so it would seem to make sense that it's limited to that number of people.
If they had lieutenants working patrol and sergeants working admin in their stead - it either shows a lack of effective leadership by those lieutenants, or by the chief. Yes, it could be that the old chief was trying to get people out of the way that did not agree with him, or, it could be that the lieutenants were ineffective leaders and could not effectively manage their people so they chose to take to the field rather than manage their personnel. Hard to say from here.

