I suspect that most Water Departments have their own lab to do frequent testing.
Some of the larger ones do some of their own testing. But there are hundreds of different contaminants per year for which testing is required and a lab must be accredited for each testing protocol used.
The water leaving the plant was likely in compliance with EPA standards. The problems arose at point sources and end user facilities. Compounds used to treat Flint river water differed in chemistry from the Detroit water chemistry.
The chemical difference stripped the plak, mineral coatings and biological buildups that were present and are present in pipes of typical water systems. In doing so, this chemistry also caused erosion of the base material of the distribution piping, most of which is at point sources.
Testing of point sources it typically done within a districts distribution system at test hydrants and not at private point sources where the most likely lead contamination occurred. These lead laterals that supply many point sources have been in place for a hundred years or more.
The question is how and when this contamination was discovered and how long the EPA sat on this information.