Would it be more convenient? No question. But, as stated, it's a philosophical question, which includes one's position on whether the two should have the same governing authority.
Any time you have two different organizations "in charge" (in this case, NCAA and NFHS), you're guaranteed to have different rules in place.
Some states have different organizations for boys and girls. Iowa, for example, has some rules variations between the two. No, comedians, I'm not talking about half-court girls basketball.
When I left the state, girls were just beginning to jump ball at the start of the game again. They moved the girls up a space on the free throw lane, keeping the bottom space open. The AP arrow always switched as soon as the thrower got the ball for the throw in. Girls coaches had the coaching box, boys coaches did not.
So, bottom line, I'm not comfortable putting the NFHS in charge of college ball, or the NCAA in charge of HS ball, or a third organization such as USA Basketball in charge of both for the sake of clearing up some minor inconveniences. Maybe it's just the federalist in me.