Being a mere observer, and not a manager, I will concur with both Steve's and Dale's comments, but also note that it is often difficult for organizations to have the sorts of slots available that will provide optimal development for new managers. Not impossible, mind you, but the primary focus is always on the business lines, and not staff development. If a perfect developmental slot happens to crop up, great, but it can often be the case that people find themselves in over their heads. My own manager is fond of telling me "Mark, you were the smart one. You didn't apply for this job."
Do we spend too little time assessing people for management positions? I can't speak to your system, but I will say that assessment of managerial candidates in our own system is much more labour-intensive than assessment of people for operational/line positions. Nobody applying to be a junior analyst will sit through the day-long assessment center that a managerial candidate does, or involve as many assessors.
But of course, spending more time and manpower at it is not necessarily the same as looking for the right things. Likewise, looking for "the right things" is also not the same as having looked for enough of the right things, or having found them in unambiguous fashion
Then, there's that whole social role thing. A few months ago I asked if it was possible to mentor "up" (i.e., provide guidance to those higher in the hierarchy than oneself), and noted that being in most sorts of leadership roles is antithetical to being seen as having weaknesses or insufficiencies. If it was easy for a new manager, and within the expectations for the social role, to ask staff "How am I doing with respect to X? Is there anything more I should be doing that I'm not?", then it would be a slam dunk to assess and pick people for those jobs, because there would be a shared understanding that the person was almost ready, and with a little filing, sanding, and buffing here and there, in situ, they could become ready. But that runs contrary to how we approach such roles, societally. We expect people in leadership roles to be completely ready, and if they're not, they cannot have that role, or we won't accept them in that role.
Then there is the unspoken principle that if one is selecting for a position, you sort of have to be smarter than the incumbent, such that you can adopt the bird's eye view of where they fit into the grand scheme of things, including future needs, and tha's a pretty rare quality. Easy to do if you're picking someone a few notches below you who will take their marching orders fro you or one of your subordinates, but harder to do if they are essentially a peer.
So, again, the 7-word descriptors are all admirable things to aim for, but getting a bullseye on what you're aiming at will be tough slogging.