Faced with a similar problem, but I have all the fun:
Time capsule 2GB, 4th Generation A1409. If i start it – it works. If i turn off and start again - it does not work:
The LED is not lights;
Ports do not lights;
No AirPort connection;
Fan, HDD – spinning OK!
If there is no power for 1-2 days, I'm trying to start – it`s works! I think this is the super magic of Apple!
I'm Check the power supply - running output of 5.2 and 12.2 volts. This unknown element "J6" - as "broken" as pictured above. In my opinion, it's just a bridge into a ferrite ring and its replacement will not solve anything. All who have such a problem, write what you have voltage at the output of the power supply.
I'm trying to connect a working power supply from an old PC. Nothing has changed - if you turn after a long wait - works. If you turn off and turn on again - it does not work! It can be concluded that my problem in the power supply and the element "J6". I think the problem is somewhere in the motherboard and the problem is connected only with the power supply circuit 5 volts.
Found here is the solution: https://sites.google.com/site/lapastenague/a-deconstruction-of-routers-and-modems/apple-time-capsule-repair/gen4-a1409-issues
But I do not try it, I think the problem is not a bad soldering or overheating, and something else.
If you see new ideas - I try to warm up this motherboard.
Interests Your ideas!