If by "internet" you mean your browser/over HTTP, then yes. But the IPFS client provides exactly that, on your local machine, without any restrictions. If you offer it to the public, you can choose your own policy on copyrighted etc. content.
I mean it's unusable to the general public since IPFS network is not accessible via the browser without a proxy.
In other words, one would first have to integrate an IPFS client into all major browsers.
No, you don't need one: you are supposed to run your own IPFS node. There are even browsers that supports IPFS out-of-the-box.