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I think you hit the nail on the head there -- "intent" is the key. If you know someone would be offended by the phrase (any phrase) and say it anyway, then it's not "the phrase" that was offensive -- it was the intention.
Another thing, kind of unrelated, is that I think "have a nice day" is so played out that when I hear someone say it, I think "ok, this person feels obligated to say something pleasant, which I appreciate, but probably isn't putting much meaning into saying it."
I work in hospitality, and I know that it's rather easy to tweak things and give big differences in results. So when it's time to say goodbye, I don't say "have a nice night" specifically because of what I think when I hear it. Instead, I change out some of the words, and make it into my new "default" (though I actually have a few that I sort of rotate so that I sound original every time). I'll say "have a wonderful evening" or "enjoy the rest of your day" or something similar. The meaning is the same, but the originality brings attention to what I say, rather than letting it fade into obscurity like the commonness of "have a nice night."
And I think many people saying "have a blessed day" have followed a similar pattern of replacing "nice" with "blessed" and similarly let it slip through their lips as their new default, not actually meaning it each and every time.