Tv stays in standby; won't turn on at the switch or with a remote

Try following the steps below.
Step 1
Press the power button once. If the TV does not turn on, observe the LED indicator, usually located on the TV's lower front panel. If the LED indicator turned off after you pressed the power button once, it is in standby mode. Press the power button once again to exit standby and turn on the TV.

Step 2
Observe the Standby light if the remote control will not work. If the Standby light stays on, the television is in Standby mode. Press the power button on the television, then try to use the remote.

Step 3
Turn the Energy Saving Switch off, then press the power button. The Energy Saving Switch is located on the rear or side panel on most Sony model TVs. Refer to your TV user manual for the exact location of the switch.

Your tv sounds fairly normal to me - there is a difference between a tv being switched off and being on standby. It is usual for a tv to need to be switched on manually and from there the remote would be able to switch it between on and standby.
Sometimes the "power" button of the remote only switches the set to standby and selecting a channel would switch it from standby...

Check the power supply board. Sounds like the problem is there.

Someone is going to have to go inside and check voltages and other things --no brand or model is indicated here so this is as much as I can suggest.

try switching on tv point remote at the tv pressing the menu/exit button and holding for 10/20 seconds hope this works for you

Powering up do you hear the relay click then you may see the display flash (not necessarily), but then the TV clicks back into standby mode. If so there is a component failure.
will need diagnosis by a repair shop. Too many different components it could be for me to guess. but
the common repair kit containes1 fuse (F302), 1 diode (D304), the power resistors (R927 and R929 for the board revisions with these two resistors) and the IC301 voltage regulator IC.

red light stays on because tv is in standby mode when switched off by the remote, should be an on/off switch on the tv body somewhere

Please check batteries in remote that seems not working. Sony TV would either turns On or standby light will blinks if TV gets any internal problem.
You question shows standby light stays On that means TV does not receive power on command from remote.
If standby light is blinking in sequence, let me know how many times it is blinking and also intimate exact model number of TV to find out possible solution.
Thanks.

have you tried changing the batteries just to elimate this problem

change them

sometimes if the batteries are low in electronic devies, they send a wek signal to its receiver (your tv),

your tv gets confused and then switches itself to standby

if you have changed the batteries and this problem still persists

it can only then be th PSU or power supply unit, take it to your local tv repair shop of mastercare somewhere like that, its going to be cheaper to get it repaired than replacing the whole tv

PSU'S generally 20 to 50 pounds depending on model

don't deo it yourself though unless you've got a capacitor discharger (or a nice 25KV will be on its way to your body)

hop this helps mate, let me know how you get on