In the Gen5 TC there is a fan at the top of the unit .. and if that fan gets old or sticky or gummed up (fans tend to do that in dusty environments).. then it will cease to be effective.
The time it will overheat is when you are actually copying files to or from the TC.
I recommend you deal with apple via applecare.. they are being unreasonable to not own up to faults.. they know full well a reset will do nothing at all.
Do you have any computer with applecare? That will cover the TC as well.
If the TC is less than 2years old.. I think most will be.. you can still buy applecare if you have a computer less than 1 year old.. or an Apple TV.
You can still buy an Apple TV.. so here is how you handle this.
If apple refuse to accept support since it is out of warranty.. and it is less than 2years old.. buy a Apple TV.. do not open it.. buy applecare for said Apple TV.. in the terms and conditions it says that the Apple airports are covered. Double check with them of course before you do this.
Be polite but assertive. Read and know the T&C.
This is my local version.
http://store.apple.com/au/product/MF219/apple-tv-applecare-protection-plan
It says.
The AppleCare Protection Plan provides local Apple hardware service options around the world, both parts and labor, from Apple-authorized technicians around the world. Apple hardware coverage includes:
- Your Apple TV
- AirPort (3)
- Included Apple Remote and power cord
And the airport bit says.
(3) AirPort must be purchased up to two years before your Apple TV purchase or during the term of your AppleCare Protection Plan coverage.
Purchase the AppleCare Protection Plan to get up to two years of expert telephone technical support and additional hardware service options from Apple. For more information, click here.
So now you can ask them to replace your faulty.. and it is faulty.. TC.
Keep the records from your iPad which will show the overheating btw..
Then ask for a refund on the unused and unopened Apple TV since you don't need it.
Therein is your problem solved.
If your TC happens to be 2years old.. which means you bought it right on release then tell us.. because you can probably fix it with a bit of DIY.
See https://www.ifixit.com/Device/Apple_Time_Capsule