Jr. is absolutely part of my official name in the USA. It is the only distinction between my and my father's name. Many forms have a specific spot for suffix.
No I'm stating that Jr. is an official part of Norman John Harman Jr., my name. It is on my birth certificate, filled out on my tax return, etc. In any case were I'm required to use my real name if I used Norman John Harman Sr. I would be committing fraud. Likewise fraud if I left off the Jr. in an attempt to confuse with or impersonate my father.
By your reasoning, there is no such person. If there can be a "Norman John Harman Sr." then there can also be a "Norman John Harman Jr." who does not have it listed on official documents.
When your father dies, and you've named your son Norman John Harman as well, don't you become senior? It's can't be an immutable part of your name if your junior/senior status changes.
To me, Americans with III after their name always look like they're pretending to be royalty. Where I live, only monarchs have that (and only after their first name).
One memorable day at work a POS Kodak system decided that it wouldn't store a particular record. Nothing worked. This happened a few times until it became clear that the only things these cases had in common was that the people's first names started BRE. I can remember a rather sad looking IT manager nodding with agreement having tried everything when I suggested we just get them to change their names. Bug is still there. It's just a historic archive now, thank god. Worst software ever.