Xbox 360 Elite Randomly Freezes

My Xbox 360 Elite recently started freezing randomly. At first, it would only happen after 1-2 hours of gameplay, but recently, it has worsened to happening roughly every 15 minutes of usage. When it freezes, it stays frozen until the power resets. Removing the HDMI does nothing. Note: There is no red ring of death.

At first, I thought this was an overheating issue, so I let the system cool down overnight, and tried again the next day. This didn't help, so I diagnosed the other common heating problems: bad ventilation, dusty components, fan issues. I disassembled the machine and cleaned out the dust, and confirmed that the fan was in fact working. I also moved the machine to a better-ventilated area, but none of these solutions seemed to work.

Also thought that maybe it was a bad copy of GTA V that I was playing. So I put in my copy of Portal 2, with the same results. Randomly freezing.

The last thing I tried was to clear the system cache from the hard drive, by going to the device preferences. This seemed to work for a short period of time, but again, the random freezing continued.

It has attempted to update itself a few times, but I am not sure if it has ever successfully completed the update.

I am led to believe that this is an issue with the hard drive. What I will try next is to use a second hard drive, and see if the machine has similar issues. But apart from that, I am stumped.

What would be the cause of this kind of issue on the Xbox 360? Again, I have a 360 Elite, with the 120 GB "external" hard drive (it mounts to the top of the device). Not sure which software it is running, but it has been updated at least once in the last year.

Any help is appreciated, thank you!