IMO Family Guy is
indeed the rightful successor to the Simpsons show, but not in a good way.
The Simpsons started getting bad when they began to go way to far into micro-second "fantasy interludes" and dropped any pretense of internal logic, realism, and character and situation driven story-lines. It quickly descended into the worst kind of drivel, and became sort of an illustrated version of a bad stand up routine from some third-rate Vegas bar.
Family Guy takes that shift and runs it out of the park with it. They also add a generous helping of cruel, nasty, negativity on top.
As dumb as some of the later episodes are, at least Homer Simpson is at the bottom of it all, a good person with a heart. There isn't a single character on Family Guy that actually is a "good person" with the exception of the dog, Brian. Even he is constantly being set-up into situations where he has to lie, cheat, or do some other nasty thing so that he won't seem like he's better than the rest of the reprobates on the show.
The "message" of Family Guy (such as it is), is that we are all shallow, cruel and selfish people and it's humour comes from a very uncool, relishing of that fact. Whomever writes it must have a particularly cruel bent, and an almost hopeless view of humanity in general. I'm betting that he used to pull wings off of flies as a child.
IMO, if you are the kind of person that laughs out loud at "America's Funniest Videos" when a baby bashes it's head on the sharp edge of a coffee table, then you will find Family Guy
hilarious.