i wouldn't jump right in to replacing the video card, it's not particularly going to be the problem you are experiencing, i just mention it because it could be. without proper diagnosing i can't say for sure. ideally, just toss it in your computer and see if it works.
both are options. with ebay, you might end up with the only parts that are available to buy are parts known for repairing arcade machines and video games... these can typically fetch 3 to 10x their value. sometimes you can find a seller just blowing out a motherboard to get rid of it. crap shoot.
its a used part, with unknown history and it could work for a day, or 20 more years.
that said, price out what it would cost to get a replacement used board... and compare it to what it would cost for a place to swap out the caps on the board you have for you...with a 50:50 chance of working... then roll the dice as to what you want to attempt.
if it doesn't have one, hook a speaker up to the motherboard to listen for any BIOS POST error beeps. if it does nothing but spin up fans, it's something with the motherboard. if it beeps out a message, go from there. the motherboard will tell you if you have a botched video card or ram or bad CPU with it's POST test beep code.