I would have said that "I happen to be busy" is usually emphatic, rather than less emphatic. I would usually expect it to be used when someone asks someone to do something, and they say "I happen to be busy" to indicate forcefully that they are busy. More than that, however, this is based on the "come by chance" meaning. It is an ironic statement - apparently saying that it's pure chance that they are busy, but meaning that it should have been obvious that they would be busy.
Of course, said in a different way, it might be literally "you caught me at a bad moment, by chance I'm busy right now". That might seem like de-emphasising, because it takes the sting out of it - you did no wrong by asking, but they're busy right now, sorry.
And said a different way, it can even be "I'm busy right now, but that doesn't matter", like "I happen to be busy, but I'll join you for coffee anyway". In that case, the "happen to be" doesn't actually contribute to that meaning, it's just there because it's a phrase.
As far as I'm concerned, it's always about chance, but it can be used to convey a lot of different meanings, or at least contribute to them. It's often used disingenuously, as well, as someone who's deliberately gone out of their way to go somewhere, to visit someone or attempt to run into them "by chance" might say:
Well, I happened to be in the area, so I thought I would pop in.
I'm not aware of any general use of the phrase to reduce emphasis.