Many sets use a mechanical relay which may be what you heard.
Based on what you have told me here with your question I agree first place to look is power supply.
All sets have as part of the power supply a standby supply that runs all the time when set is plugged in---it supplies the voltages for the remote or manual on and off to power the main supply and with it the set to be turned on.
You are able to safely get the back off---lay set face down so entire screen is supported evenly---no downward pressure as the screen unit can be instantly destroyed.
Most sets have to be taken apart this way if a base is attached as it will have to come off to get all the cover screws off.
You may be on the right track with filter caps---that standby runs them all the time and the material in cheap caps can dry out in less than two years--using cheap ones cuts cost when making the set.
I will attach a few photos of what to look for---one is a mild example and another has some extreme examples---if locate one or more I can explain how to replace and identify their values--not hard as each have only two leads soldered in--values are actually marked on the parts--each has a plus on one side and a minus on the other.
let me know if you find bad ones.
look for perfectly flat tops like on left===right one is just slightly puffed up and is still bad.
The ones with arrows are extreme examples and some of the others look ok.