I read the post regarding digital cable boxes rebooting and showing error messages if the boxes get too hot, and was amazed. It should not matter whether the TV the box is connected to is tube or solid-state; the box should work just as well with either type.
Also, I honestly don't believe your cable operator would "freak out" if one of their technicians saw a digital STB hooked up to an older tube-type television; at least they shouldn't, as should be none of their business how old a subscriber's set is. Take Doug, drh4683, as an example. He has TVs in his home anywhere from 30 to sixty years old, and probably has at least several of them connected to cable at any given time. I bet the cable company doesn't bat an eye.
All makes of digital cable boxes, again, are designed to work with any type of TV made, so if a cable tech saw what you described it would be, to say the least, unprofessional behavior on his or her part. These days, older TVs, to say nothing of late-model sets without digital tuners, absolutely need a cable or satellite connection with a cable box to work at all, no thanks to the DTV system the FCC forced on us last June.
There is one case where any television ever made in the last 50 years will work on a digital cable system without a box, however, although with that particular hookup all you ordinarily get, at best, is expanded basic cable which, at least in my area (I have Time Warner cable) runs from channel 2 to channel 99 or so. Anything higher than channel 99, of course, is digital, and you need a box to get those channels. I have TW's basic (what I call "bare bones" basic) cable which is broadcast channels only, plus WGN America and a very small handful of special-interest and public-affairs channels. I do not have or need a cable box with this service, although I did have digital cable several years ago. The box, a Motorola metal-cased unit which was built like a tank, ran warm, but not so hot as to burn one's hands or fingers. I'd still have digital cable today if the rates hadn't gone sky-high about five years ago. Crying out loud, even expanded basic now costs $55.67 a month, which was much too high for me as I live on a fixed income. I had my service downgraded to basic a couple of years ago and now use my VCR and DVD player for programming I used to get on cable. Same level of enjoyment (even more, really, since I can pick and choose what I watch with a VCR and DVD, as I have a rather large library of programs in both formats), but a lot less strain on the wallet.
