First point, nobody on this site is going to give you a legal green light on doing this. This is not the place for legal advice, also this is not even borderline.
It is public information that Youtube had plenty of troubles with copyright so don't assume it was smooth for them and they were without problems. Their argument was that content is provided by users not themselves, you can't use this argument because you are providing the songs.
Also to point out the example provided by you is not how Youtube gets around copyright. To my knowledge they now have deals in place with copyright holders and when a song is identified then they deal with that song according to instructions from the record company, this could be placing advertising etc.
If you were to place copyright MP3s on your site you are definitely putting yourself at large risk and I wouldn't even consider starting a business based on that idea. You could find yourself having to compensate a record company for the number of downloads you are assumed to have created. This is not even semi-hidden like peer-to-peer and you will find this likely violates your hosting agreement.
One area that you could explore is having embedded Youtube videos, I don't know what Youtube's agreement says about this, but it is one option you could look at. You could also provide the songs lyrics in combination, but again, the lyrics are also copyright material.
If it were me and I was using it purely to educate and make the world a better place and just trying to make a little advertising money on the side I would try and look into find some free for commercial use songs. I know there are some sites that license modern music under creative commons license.