I am not qualified to give you this answer - I know very little about TV sets...
If the switch makes the TV come on for however short a period then the switch is working.
Switches Never have fuses to protect them - and if the set turns on even if your set did have a fuse-protected switch then the fuse must be good - if a fuse - any fuse had blown then the set would not turn on at all.
I would guess that all the start-up circuits are working well but that the power supply has a fault and is automatically switching off.
It will probably need a specialist to solve the problem.
Good luck...
Peccavi
March 2011