Okay, there are 2 or 3 approaches that might help me clarify my thoughts:
To have done something, taken as a grammatical construct, sounds more like an act of achievement, completion, finishing, perfection, etc. The negation of it would be not to have done the thing, rather than to have not done the thing, which would be having nothing completed.... I know it sounds rather silly, but it seems far less sensible a declaration than not having completed something. It makes me think of Douglas Adams` prose in The Hitch-hikers guide to the Galaxy.`
Au contraire. By your own argument, it is more sensible to refer to positive occurrences. If there has been a failure to complete a task, that is the positive consequence; it's what we have:
Having (not completed the task) - a positive statement of what currently obtains.
Not (having completed the task) - a negative statement.
Not having (completed the task) - another negative statement.
The second strand, and the stronger one in my view, (I'd hope so! ;-)) is made clearer by taking a step back and comparing the structure with being, which can be taken as grammatically analogous.
Would we say....
Not being a doctor, I wouldn`t want to give a prognosis on her condition.
.... or, ....
Being not a doctor, I wouldn`t want to give a prognosis on her condition.
Either you have an archaic and unnecessary inversion here, or a malapropism, if you ask me.
The mistake you've made here is to use "be" as the main verb. This same mistake can be illustrated, more relevantly, with "have".
1. Not having a car, I couldn't drive to work.
2. * Having not a car, ...
Even so, it's quite normal to say
3. Having no car, ...
However, in the question at hand, "have" is a modal verb. The argument is "not having <main verb>" v. "having not <main verb>
The third argument I suppose would be centuries of written English which exclude the second answer. Having not completed my homework is probably nearly as normatively `new` or `urban` as `me and him both agree.` You hear it, but it`s not the norm from the grammars.
Can you cite one of these grammars?
I can't accept on face value that, "Having not completed" is new. I'm willing to find examples from literature if necessary.
Hopefully this is an okay explanation of what I was thinking.
So, point 1. seems right, point 2. is about a different structure, and point 3. is assertion? I'll accept that that's what you were thinking.