This is controversial. Some studies have found that petty crime like this leads to more crime. Or stated differently if you solve these little crimes there is less big crime as well. However this is very controversial and those studies have been criticized - I am not able to figure out the truth here, feel free to do your own research.
A policing tactic back in the 1990s was 'Broken Windows', which did what you said - fix the little stuff and the big issues will improve. It was controversial, as you say, because at least in some places the strategy became 'oppress black and brown males' so the white people can feel safe.
I've read one news story on an analysis of Broken Windows (so not a lot of data) said that outcomes were not correlated with that strategy. Crime went down everywhere, whether or not they used Broken Windows.
Mayors and police chiefs, etc. are called geniuses or fools in strong correlation with national trends, especially the economy. Lots of 1990s mayors, etc. were geniuses as the economy boomed and crime came down nationwide.
