Can a LCD display work with swollen capacitors for a large period of time until briefly turned off?
Without seeing the schematics of the TV, this can only be speculation, but - yes, it is possible for some damaged components to allow the television to continue to function but prevent it from starting up.  Startup is a very different set of conditions for a power supply (and other subsystems) than regular operation.

Of course, capacitor plague is far from the only thing that might have gone wrong with a 15 year old piece of consumer electronics.  And many of the other potential failures are impractical to diagnose without special equipment and knowledge of the design.