Answer to "A red light comes on but tv will not power on"Your main board(HDMI, usb component connections etc… area) must be replaced. A used one can be found on ebay for 45$ or so. It can be replaced fairly easily via YouTube tutorials.
Answer to "Why doesn't my RCA HDTV power on"If you are receiving a stand-by light ( red led light when plugged in), the TV's power supply board is (PCB) is still providing power to the set but your main board ( USB, HDMI, Componant cable connection area) is not functioning properly and will require replacment. If you are not able to see the stand-by light then your PCB is faulty or can no longer provide the require power to the main boards nd must be replaced. EBAY has both boards for sales and as well youtube has some simple tutorials. Tampa Tech is a good all around informant and DIY channel for tv diagnosis. Hope this helps.