There's a lot to unpack here. tl;dr is "use your best judgment" but that's probably not very helpful. So, in no particular order:
For one thing, the message is just plain old.
I think it's as old as the mod analytics page itself, and I don't think it's ever been changed, even as the analytics themselves have been updated (one random example) and the site-analytics privilege was introduced.
This message isn't as much about the contents of the analytics as what people do with them.
The more the analytics get shared, the more people spend their time navel-gazing and discussing the pretty graphs and numbers instead of actually asking and answering questions about [insert site topic here]. We've known for a long time that it's healthy to have a meta site where people can do the work of building and caring for the site itself, but it can get really easy to lose focus on the topic at hand and start focusing more on meta-issues like procedures and close vote rates and analytics graphs than the main site, which is also bad. Asking people not to share the analytics just preempts that problem in the first place.
On top of that, the analytics can be misleading. Sometimes they require or benefit from more context than they're presented with, and cause more trouble than they're worth when released, even if they're shared with good intentions. Again, not a problem if they're just not shared in the first place.
Availability and accessibility are not binary.
Several of your points have to do with the analytics info being available in other places. There's a difference between something being theoretically publicly available and something being advertised. It may be possible to get some of the analytics info from other sources, but it generally takes more work or a certain level of access to do so than it does to just load up the analytics page. If it's harder to do, people probably aren't going to do it as much. And facilitating that kind of data analysis isn't really our goal here.
It's good for the mods, and that's good for us all.
Mods really do have a special place in our system. They do a lot of thankless work motivated by nothing more than love of their sites (and receive as payment only occasional written abuse from trolls). More to the point, they're uniquely positioned to know what's going on in their communities. We on the team rely on them as our primary liaisons with the various sites. They are more capable of interpreting the data, and have more use for the data, than the average community member.
To be perfectly frank, it's nice that it feels a bit like a special perk, too. This was probably more true before the /site-analytics feature came out, but it still applies.
To answer your question, then... I guess maybe a better banner message would be
the analytics information on this page is provided to you because you have a unique need for it and ability to use it; you're not prohibited from sharing it, but please try to limit your sharing to times when it's really necessary or impactful, so that it doesn't distract a lot from the primary business of the site (which is Q&A) or reduce the impact of seeing analytics; and if and when you do share, do your best to provide a lot of context so that you don't inadvertently cause more trouble than benefit
but the current message is a bit snappier, so consider it a shorthand for the above.
