A few notes from my experience in Center Point:
-They had/have a requirement for separate business licenses (owners, PMs, and contractors), making it harder to start work in Center Point.
-They did/do annual inspections (similar to Section 8), where they pick apart maintenance and usually come up with additional fixes.
-Very hardcore about permitting. If they drive past a worksite they'll stop in and ensure the permits are in place.
-Have heard horror stories of the city requiring that multiple points in the home be brought up to code, even if you are only working on one. Ex: HVAC replacement during rehab/reno, they then require plumbing and roof and other stuff to be brought to new code.
**All this to say that there was a class action lawsuit (by owners and PMs) initiated sometime late last year that I believe is still in process. I do know that the initial result was a restraining order on the inspections and requirement for business licenses. Not sure if that's changed by now, but at least for some time they could not require those.
Just like many areas in Jefferson County, it's a street-by-street neighborhood. I'd say the majority of it is fine, but definitely want to drive the streets surrounding your target property to get that true feel.