Which components should be checked on a faulty tv power supply?
I don't have the 5V stand by to the main board and by looking at the
power board's schematics I should have some 12V lanes as well to the
main board which I don't have either.

Assuming you have ~400 V on the main power supply filter caps (C116-C121)...
This suggests the standby power supply isn't working. The controller is IC303, an ICE3BR1765J, which has an integrated high voltage MOSFET and a 'startup cell' which takes voltage off the FET's Drain and drops it down to the lower voltage which is required to get it running. Once it has started up it powers itself via the 'AUX_12V' winding on the transformer.

First check the value of current sense resistors R301-R304, which should have a combined resistance of 0.5 Ω. If these resistors are open circuit then the IC is fried, and both it and the resistors need to be replaced. If the resistors are OK then measure voltage between pin 5 (MOSFET Drain) and pin 8 (HV ground). Note that both these points are at a high AC voltage, so be careful! If you see ~400 V then the IC is probably faulty. If you don't see high voltage on pin 5 then power is not getting to the IC for some reason, perhaps a 'dry' (cracked) solder joint or broken track on the PCB.
Other causes are possible too, but these are the most likely ones. The controller IC has high voltage stress on it and is susceptible to mains spikes. Sometimes when they blow up they take out other components around them, so you should also check the optocoupler (IC301) etc. before replacing the IC.