I know a lot of people see Johnny Davis as having a high floor -- but I disagree. I also don't see the comparison to White. White has always had a league average-ish true shooting percentage of around 56%. At Colorado, White's true shooting percentage was 63.5%. Davis's true shooting percentage last season at Wisconsin was 52.3%. Furthermore, White is a strong playmaker and passer, while Davis is non-playmaker and a poor passer at this point.
If Davis doesn't become more efficient, I don't think there's enough else in his game for him to lean on -- at least at this point in his development. He's a good defender and great rebounder for a 6-foot-6 perimeter player but I don't think that's enough for him to remain in the league without improvements elsewhere.
Like you alluded to, Davis might not be able to get separation in the NBA. If that results in him forcing even tougher shots than he attempted at Wisconsin, that true shooting percentage could plummet even more.
The hope is that he is able to utilize the added space in the NBA and the improved talent around him to fine tune his shot selection. I'm reasonably confident that will happen, which is why I'd fine with the Spurs taking him at 9.