If you are looking for historic accuracy I Dream of Jeannie (and all other shows of the era up until 1998) were always broadcast in 480p-SD. Any HD versions you might find are a misguided modification to the original format. The same being the colorized Season 1 (put out later by Sony) which was originally broadcast only in B&W.
Cats4U
Quick, pedantic clarification: I'm using the term 480p as it's the 'settled upon' term for industry standard, pre-HD video format - not because it's actually an accurate representation of the actual resolution used by TVs in the past. I only say this because when I used to work in the industry, US TV was often referred to as having 576 lines, and I never bothered to do any research in an effort to understand the discrepancy. I'm also using several other terms rather loosely here, so you'll notice I surround some terms with single quotes - this indicates that I'm probably over-simplifying (while trying not to effect overall accuracy) in order to keep this from being even longer. With that said, on to my actual, somewhat rambling reply...
I want to be clear that I'm not saying your take on this is invalid - just that my take is different.
TLDR: Anything not originally shot on film, and/or not originally shot in digital HD/FHD/UHD/etc. would have to be upscaled - but pretty much any material that was originally shot on film (with the film still being available as a mastering source) can be remastered in HD+ without any modification (see below) or upscaling - and that includes most TV shows all the way up to the 90's at least. Colorization is definitely modification, but whether or not it's bad is a matter of opinion - it could be argued that colorization allows the material to viewed as it was originally intended to be, and it was only created in B&W because of the limitations of the time. Clearly this isn't true of all B&W material, but much of it was only in B&W due to technical limitations (early stuff before color came to be) or cost.
Long version:
I'm not sure what you mean by 'historic accuracy', so please forgive me if I'm telling you things you already know and have missed your point entirely. Shows like 'I Dream of Jeannie' were *broadcast* in 480p, but almost none of them were shot in 480p - shows from that era were typically shot on film, which is analog as opposed to digital, so the only resolution limit is defined by the actual physical structure of the film material itself - and even the lowest quality, grainiest film used by the industry is typically capable of producing a picture of much higher resolution than 480p without any kind of upscaling, modification, or visual quality loss. Remastering older shows in HD (or even 4K) is not a 'misguided modification to the original format', it is simply making use of the massive amount of extra picture information that is available via the film (the original format) but not usable via the 480p transmission/viewing format that was standard at the time - no 'modification' necessary. There is a slight caveat - remastering special effects that were created in 480p and composited *after* the filmed material had been downscaled to 480p, such as what was done with the original SD/480p airings and DVDs of Star Trek: TNG, DS9, and Voyager obviously involved 'modification' in that all the special effects had to be redone from scratch. Shows, like Babylon 5, that had all the live action shot on SD/480p video in addition to doing all the effects in SD as well present even more challenges and their HD versions could fairly be called a 'modification' since the material would actually have to be upscaled, resulting in a picture that consists of image data that did not originally exist in the source. As already stated, the standard TV format, prior to HD, was 480p (the 'studio' format was slightly larger, but the extra image information, called over-scan, on the top/bottom/sides was not recorded for broadcast and/or transfer to 'consumer' video tape) - any material not already shot in a 480p (e.g. shot on film) would need to be 'downscaled' to fit the standard 480p video format. The reason things not shot on film were shot at 480p was because that was the standard and when the standards were set there was no consideration of a future with higher resolution (or things like DVDs, BluRays, etc.), so there was no reason to make video equipment capable of shooting at a resolution higher than what TVs could display as it would then have had to be downscaled to 480p anyway. Film is why so many pre-digital movies & TV shows are available in HD, and the difference between the 'final' cut of a film and the 'final' cut of a typical pre-HD TV show explain why HD versions of movies often came to market so quickly and HD versions of SD TV shows took longer. Usually the 'final' cut of a film is just that - a set of reels of film, that when shown in order represent the full film story. Because of that, creating an HD+ version of a film is often trivial compared to doing the same for a TV show. A TV show's 'final' cut (at least from the mid '70's onward, probably earlier) was typically on 480p video, which meant that upscaling involved going back to the original cans of film, finding all the shots that were actually used, converting all that to digital, and re-editing/re-assembling each episode from the cut sheets (if those are even still available). Any special effects not done in/on the film itself were typically done in 480p, so they would all have to be recreated entirely from scratch. Then there's also the audio, but that's an entirely different ball of wax.
So, now having rambled for a while, getting back to what you said - unless you're talking about something I'm totally missing - HD versions of shows like "I Dream of Jeannie" are not modifications, they are actually truer to the original intent than what was shown in pre-HD days (which required them to be 'downscaled'). That could even be said about colorization since subsequent seasons were in color, and if B&W had been part of a show's character, then it probably wouldn't have changed to color for subsequent seasons.
