Look at the SEC and what good their regulation did. They totally ignored Bernie Madoff (under Bush) and Enron (under Clinton), giving regular folks a false sense of security in the market. If there was no SEC, people wouldn't have a default assumption that the market isn't rigged and they would invest more carefully.
I really don't like comments like this as they are completely unproductive. Why, Fred over there got robbed for all the good the laws and cops did! Guess we shouldn't have any laws or cops at all, giving the folks a false sense of security that they can leave their homes without being armed to the teeth.
*sigh* The idea is not to abolish something when it fails, the idea is to see where something failed and improve upon it.
The answer isn't to regulate the internet, it's to get rid of the whole monopoly provider system. Have a regulated (even non-profit) independent company (can't be owned by an ISP) run and maintain the network, deriving its revenue from the ISPs wishing to use it.
This fails in a number of ways. First of all, you are just replacing a bunch of local monopolies with one big centralized one. With your suggestion of just regulating THEM, you end up really regulating the Internet. Welcome back to square one. Worse yet, your "centralized non-profit" would likely be a Government Sponsored Enterprise. As you say about the Health Department, GSEs don't have a great track record of providing great service, because they have little motivation to do so.
Lastly, the end-point "provider" companies in your scheme would struggle to find some way to differentiate their product from others. Price can only go so far, so then you'll get into network segmentation, walled gardens, "premium content", etc.... Net Neutrality effectively done for.
So, with your scheme you get the worst of both worlds -- you get a huge centralized (and probably government run) monopoly AND no net neutrality to boot.