I think the older post notices can be hidden from individual sites using a little CSS:
However, I think after a certain window of time these notices become relatively meaningless and serve only as extra noise, which I think applies to all sites. It would be useful if these notices could tidy themselves up after their usefulness had passed.
Maybe it would be a good idea to just have .post-notice.old: display: none; shipped in the default CSS so people can always override it if they want to. That makes the change very small and it leverages the existing .old functionality.
The CSS in page code doesn’t appear to differentiate new/returning user notices from staff notices, though these are very different types of notices. So, as near as I can tell, suppressing old notices via CSS suppresses all of them.