Problems with games and hard disk on Xbox 360 S

Hello, this may be a bit long but the issue here is quite complex and I want to make everything as clear as possible by giving all the details needed. Please, if you are a Xbox 360 expert, give this a read.

A few days ago I was playing "Arkham City" and it started showing problems. All of a sudden, the infamous "Disk unreadable" message showed up, taking me back to the dashboard. I loaded the game again and it seemed to work fine, but some days later the voice
clips would come in late and after some minutes the error would show up again. I looked it up on the internet and found out it is a common problem which was apparently never really solved.

Trying to restart the console, I noticed that the logo screen would stay on suspiciously longer than usual, so I went to check the hard disk and noticed it took very long to load and some games were showed as corrupted at random, with the portrait with the
white exclamation mark in it, every time a video game was inserted. I had a hunch and tried other games, namely "Arkham Origins" and "Arkham Asylum" and after some time, they showed the error message, too.

I tried another video game and it soon began to lag heavily, like it never did before, until it froze. At this point, the batman games began to also randomly give the other infamous error message, "Damaged downloadable content detected", in addition to the
previous one. Other video games without any downloadable content in them work just fine.

I tried, in order, to empty the system cache, to reset the console by using the LB RB X LB RB X combination and then update it and to play without selecting the hard disk as a storage unit, using a USB key if I wanted to save. Now, the first solutions were
completely useless but by playing without the hard disk selected I was able to play much longer but the message would show up nonetheless, albeit much later.

I phisically removed the hard drive and, what do you know, now I seemingly can play as much as I want without the message ever showing up. The voice clips still come in late but rarely and only heavy loadings, probably because the game is obviously not installed.
Could the hard disk be the source of the problem? Will formatting it solve everything or that would be useless and I would do better just buying a new one?

:)