There seems to be QUITE a bit of confusion here...
First on Component vs. HDMI. HDCP is not an issue with the Apple TV. Everything you can watch with HDMI you can also watch over Component Video.
If your TV has Component Video (Red/Blue Green) as shown above, and you connect a cable from those jacks on your TV to the same color jacks on your Apple TV - PLUS the Red/White Audio cable (or Digital AUdio cable), you should be able to get a picture.
Since you have used the APple TV with an HDMI connector, I suspect you may have selected a higher resolution (say 1080i or 1080p) that your TV does not support over component video. This might cause you to lose the picture.
Not sure how to get you back to the menu on the TV, so you can select the appropriate resolution.
I don't think the issues is really whether or not you have an HD TV, but whether or not you have component video (or HDMI). There are a number of TV's that are NOT HD that have component video, and I would expect Apple TV to work just fine.
It is doubtful that your $100 (!) cables are the problem. Remember there is nothing special about component video cables. In a pinch you can use ANY RCA-to-RCA cable, just match up the connections from the Apple TV to the TV. This may be more confusing than it is worth, but if you have extra cables (even two extra sets of red/white audio cables), you can rule out your cable connection.
I think the answer may be to get the remote key sequence to reset the vide to the default (480i? 480P?)...