In the sense that aesthetics are subjective, you’re welcome to your opinion. My opinion differs.
Looking at the sentence from a phrasing and pacing context, it is jarring. The first phrase comes along
“That’s one small step for man,…”
That’s one small step for mankind, okay, he paused, what’s next?
“… one giant leap for mankind.”
Erm, what? How can it simultaneously be a small step and and a giant leap for mankind?
The first phrase sets the expectation. It defines the scope. The contrast of “small step” and “giant leap” set up the need for a contrast in the object. The parallel phrasing of “man” and “mankind” is much more poetic than using “humanity”, but it only works if you contrast meanings, i.e. use “man” to mean one person, Neil himself. He could have tried
“That’s one small step for a man, one giant leap for man.”
Using the singular “man” as the generic for humanity in the second part, while technically passible, would have been awkward. Mankind is smoother. He could have tried
“That’s one small step for a man, one giant leap for all men.”
But that would have sounded too sexist, almost deliberately exclusive of women. Mankind is more inclusive, though perhaps still a bit male centric to the uberfeminist.
Seeing as how he was talking explicitly about the small step he made off the lander foot to the dirt, and seeing how he was contrasting his small step to the cummulative effort of the nation that made taking a step onto the lunar regolith possible and what it implies for humanity’s future, it just really does need the “a” in there.
But that’s my personal overanalysis of how the aesthetics of the phrase works. What I can say from my personal experience, I recall being a child and being told or hearing the recording of what he said. Note I was not alive in 1969, so this was all well after the fact. I just recall hearing the “for man” and then the proceeding to “for mankind” and getting confused. It took deliberate effort to back correct the meaning of the first part of the phrase.
While part of me has a preference for what my ears actually hear, my overall sympathies lie with the version that is actually coherent and artistically flowing.