I don't think that is a good idea.
There is no reason to show warnings for any backend admins as flash message after every login, just because the reports module says there is something wrong.
There are other solution to keep track of important things like the caretaker.
Furthermore, I do not want to nag an admin with this. The aboutmodule shows such a warning, and that should be enough. If an admin doesn't care, that won't change, even if we nag him more.
Additionally, I really do not think that another 'nag-area' would help, and it wouldn't be nice codewise as well. Instead, we need more separation: currently, we have aboutmodules telling things, we have reports telling things, the install tool points out things, the install process does, the bootstrap does ... this is already bloated heavily, and we should streamline instead, giving different interest levels according information. (eg. install tool should only show php / system problems, reports only typo3 setup problems, and aboutmodules just links to reports if it found some problem).