Well, there is the government's FHWA study. There are a couple mechanisms.

The first is that sometimes running a red light is safer. People normally don't think about red light cameras and thus their impact on behavior only comes into play when after the driver is already in a position to get caught. This leads to people making unsafe stops when it would be otherwise more advisable to just go through the light. The link above shows an increase in rear-end collisions.

More relevant to the original point is how the city reacts. The city council enjoys money and shortens yellow light timings or blocks lengthening of yellow light timings, causing more crashes.

Looking briefly, they're all before-after correlational studies (longitudinal). These are not as good as randomized experiments, but they're still much better than a cross-sectional correlation (eg. "we looked at all traffic lights; ones with cameras have higher accident rates p=0.xyz").

For example, given a cross-sectional correlation result like that, there's a very easy retort: "people only install cameras at dangerous intersections!" The longitudinal design deals with that: "but they weren't so dangerous before the cameras were installed!"

Now a critic must look to less likely explanations: "maybe there has been a traffic-crime wave whose early phases caused both the installation and later increased traffic rates" (or something like that, I don't know much about the issue). It is to deal with all these more exotic variants that one wants to step up a level and add randomization.