I guess you're trying to be pedantic, but that's never useful when talking about humans, especially human language. And even if we tried to remove the human element, the transliteration is clearly not the same or even an accurate reflection; if it were, then transliterating to Japanese and back would get you the same word and this wouldn’t even be a problem.
The key factor you seem to be missing is that most of these words are themselves very lossy transliterations of original English words. Some become so ingrained in Japanese that they’re considered Japanese words by most people, and would make sense to leave in their lossy, katakana form even if romanized. As many people have mentioned here, “katsu” is ironically one of those words.
However, words like katsu are the exception and most are still considered English (or other language) words with the katakana being a best-effort representation in Japanese. The links shared by Zarel[0] and numpad0[1] demonstrate this perfectly.
The “original” word being expressed is “home” and “ホーム” is just the best you can do in an official Japanese script. “Ho-mu” isn’t useful for anyone, foreign or Japanese. It’s meaningless to non-Japanese speakers who aren’t familiar with the lossy transliteration to Japanese they need mentally reverse it. Japanese speakers could work it out quickly enough, but it would take more time and confusion since it’s neither representation they’re familiar with everyday, the original word “home” and the expected Japanese representation “ホーム.”
[0] https://magazine.jp.square-enix.com/biggangan/
[1] https://ja.wikipedia.org/wiki/タミル語