What's the purpose of DA selecting Non-KD complete NCOs?

First, the purpose is Force Generating assignments like Recruiter and Drill is not to be a broadening assignment for NCOs who are looking to try something new. The purpose is to fill the Force Generating mission, which is significantly more important than optimizing a Soldiers KD time. Each branch receives a mission they MUST fill and they choose the most qualified Soldiers available to fill those spots. Are we really going to tell the Army, sorry we can't recruit or train the soldiers we need to because these Soldiers haven't been in a leadership position for a specific amount of time? Absolutely not. Force generation is one of the single most important missions in the Army. If you don't have enough Soldiers then you don't have an Army. That's why your KD time doesn't matter.

Second you may actually want to find out what KD is and how much you need. As an NCO who will be getting looked at for SFC, you need to be familiar with your DA Pam 600-25, your CMFs board AARs, and your career map. You would see how much KD time you actually need and more importantly, why you need KD time.

SFAB does not set the standards for KD time. Your MOS Proponent determines how much KD time you require at your grade and in which positions. SFAB is a voluntary organization, and so they can set whatever KD requirements they want from an applicant to fill a position. Ranger School requires Instructors to fill one year of KD time to evaluate a student filling that graded leadership position.

Key Development time is an officer term. For enlisted its called Operational Experience time, but people still call it KD. KD time is the experience time your proponent determines that it takes you to learn and be qualified and competent in your standards of grade. Your standards of grade are also outlined in your DA Pam 600-25. KD time is not the amount of time you need before you can apply to SFAB. SFAB does not determine what is KD time, your proponent does. When the evaluation board meets for promotion, the proponent briefs the voting members on the DA Pam 600-25 and what counts for KD time. He (Infantry proponent is a He) briefs them that RATED time qualifies as KD time. You can see this in the promotion board AARs.
I'm not sure why your anecdotal evidence just wandered all over the place. Your KD time has nothing to do with CA or Psyops and they have nothing to do with SFAB. If you wanted to go to Psyops or CA you have had years to go already. There's no reason for your branch manager to think that you were planning on going this late in your career when you've had the last 5-8 years to apply. But if you are actually serious about applying to those, waivers do exist for exceptional candidates. Especially if you're less than 12 years time in service

KD, last I knew, was Key Development. These are the roles that the Army believes are important for success at the next level. However, as far as I know, this is not a hard-and-fast requirement. It is more of a very strong recommendation. There are paths to success (and to promotion) that do NOT check all of the KD boxes - but those paths are few and far between.

As for SFAB/CA/PSYOP eligibility, etc., the Army does not really care about that on an individual level. The Army does not make personnel decisions, assignments, or selections based on what this assignment would make you eligible or ineligible for, they do so based on the needs of the Army and an individual's ability to fill that need.

Your 12 months of KD time is the bare minimum. You can be qualified or highly qualified in the eyes of a board, just not most qualified. This is why we have KD stabilization now, to keep NCOs in KD positions. However, the Infantry are required to maintain X# of drills so people get tagged early. My recommendation to you is to be extremely proactive with your career with at drill. A lot of people get on drill and do the job, which is great, but they dont do anything to set themselves apart from their peers. Everyone on drills NCOERs are very similar. So go to boards and schools to set yourself apart and make your NCOER different.
Looking forward, while SFAB will take you with 1 year KD time, it doesn't count towards it. At least until branch updates the 600-25. If you do CA/Psyops Infantry KD time will be irrelevant.