Check the contacts for the power supply and for the power button for corosion. Since the former gets human sweat and the latter may suffer from oxidation as they get quite warm thats a good start.
I suspect that perhaps the reason for the blown caps is the reason why the TV won't start. The caps may have been a symptom for something else blowing or a power surge as theres no good reason why 3 would blow at the same time that im aware of.