OK, I'm pretty sure I know what's wrong.
You almost certainly have two IC's in the power supply that need to be replaced. Only one is causing a problem, but they should both be replaced. You may be able to make the set come on if you try the power button enough times, that's how the IC's behave.
If you have electronics experience and can solder and unsolder electronics components, you might be able to do this yourself, but it's a fairly daunting challenge for the uninitiated.
This set has them located on a double sided PC board (the D Board) and it is much more difficult to service than a single sided PC board. You need specialized desoldering equipment to remove the IC's, and frankly, many otherwise so-so TV shops can't handle this. It's the rare servicer who has the ability to do this with care and skill.
When I encounter this board, I have to take the board away from the customer's home and work on it at my shop where I can do this procedure. Ballpark cost would be $350, and that's a little higher than it might be if it was a single sided PC board, because this board is so much more difficult to work on.
But the parts are IC6501 and IC8002, located on the D Board, and the generic part number is XXXXX You can order it by that part number from MCM Electronics. If you want the Sony part number, let me know.
One other possibility for you is to contact PTS Corp. and see if they can rebuild the board for you.
Hope that helps, let me know if you'd like more info on this, and thanks for using JustAnswer!