It is always possible there is another fault.. to shut down after a minute.. if you are relating that correctly.. is unusual.. I would power the board from 5v 2A supply.. that will quickly show if the issue is your power supply or not.
The TC is definitely in need of more cooling but I would hate to suggest doing a major modification until it runs out of warranty. It is possible to improve the existing cooling with a bit of smarts.. a laptop (or netbook size) cooler to which you can seal the TC.. this is the tricky bit.. you need to allow a space underneath though.. perhaps 10mm.. and seal just the edge of the TC so it allows the cooler to draw air through the TC. That will work without modifying it. But you will need to make up a proper spacer and gasket.
I see pictures of various fans mounted on the top.. they are next to useless. Cooling the outside plastic will achieve only the tiniest improvement in heat transfer due to the thick plastic (insulating) top.. if it was made out of metal maybe it would do something.
Sounds like what is also needed is a different heat sensor that would start the fan at a lower temperature.
It is not the sensor that is the issue. The TC includes a proper fan controller. It is simply set too high.. and so it doesn't turn on.
If you want to break into the TC.. you can solder in a console.. and use serial to USB adapter.. these are very cheap.. the soldering in is tricky.
See the bottom of this page.
http://sites.google.com/site/lapastenague/a-deconstruction-of-routers-and-modems /apple-time-capsule-repair/new-issue-with-a1355-gen-3-tc
The envstat.. will show temp.
This command will turn on the fan.
i2cctl w 0x18 0x00 11010100