I don’t agree with the premise of your question. “Happen to” can be used to soften a disagreement but it doesn’t necessarily do so in your example. “I happen to be busy” (especially depending on the tone used) could convey that “I am annoyed that you assumed I was going to be free, and I’m not”. Whereas a simple “Sorry, I’m busy” is pretty neutral. “I hate that politician. Everything he says infuriates me.” “I happen to think he makes some good points.” (A soft way to disagree. It will surprise you that I disagree with you because we usually have a lot in common)

Basically it comes from how you explain to someone why you can't do something. For example: Person A "Can you help me move?" Person B "Im busy" In this instance Person B seems like he is being abrupt and not even considering helping person A. Which makes person B sound like a selfish person. Now try this. Person A "Can you help me move?" Person B "I just happen to be busy studying for my exam or I would." Its odd, but it just kind of eases the conversation a bit so its not as awkward. It also can be used to convey surprise. Person B By chance had to study the same day person A had to move, so person B couldn't help Person A. "I just happened to be at the park at the same time as you." This one more depicts the "By chance" Definition you referenced.