The power supply may be defective or damaged. In short-circuited-component cases, the power supply may still be damaged even when the shorted board at most fault is replaced. So very often either two, or three even, of the boards must be replaced.
However, if the 'shorted and not staying On' behavior continues, the shorted component is either not replaced, or still connected to the power supply and draining power from it to excess.
And no, I shouldn't pretend that these sort of cases are easy. They aren't. But they can be gotten through, with an intelligent-enough approach.
One very crude approach to diagnosis is to unplug everything from the power supply, pretty much, and see if it stays On, then connect one board up, wait a bit to see how that affects things, then another board, and so forth until one board, connected up, causes the TV to shut Off. I don't recommend this... but if other approaches have failed, it may provide an indication.
Have you examined all boards on the TV quite thoroughly, component-by-component? visual inspection cannot find everything of course, but a burnt-out component will have a brown spot on the other side of the circuit board, or will look melted/heated a bit; also, capacitors that are leaking, or the top is bulging, need replacing beyond question.