crts care plenty, especially crt tv's which almost never had scalers.
the only way this works is if the set top box can be set to output analog sd output. nothing else.
You'll still see benefit to viewing a HD signal on a SD TV - it will look nicer than an analog OTA signal. I downconverted via SVideo to a SD TV for a little while and I always thought the HD signal, downconverted to SD, was far superior than an NTSC SD signal.
Note: 1080p downconverted o sVideo tbeing compared to NTSC OTA broadcast SD, not DVD 480p "SD" or whatnot.
Also a 36" TV SD signal will still look fine provided you have a good cable provider. You are now looking for a 47" and you are going to see a world of difference on a screen like that, to put it mildly.
But, Red.. If you watch older shows such as growing pains or home improvement, it looks much better on a normal (640x480?) CRT TV.
Its not so bad, 8 feet back from a big tv it looks ok, it isn't going to be pristene, but it is tolerable, in the way you tolerate youtube video.
The only intolerable is analog input from say an old dvd player or vhs, then it looks like youtube.