Drew,

Online users do not have the ability to mute hashtags as the email
subscribers do.

They do, but not as conveniently:
https://groups.io/g/GroupManagersForum/wiki/Help-Mock-up#Managing-muted-and-followed-hashtags

They aren't able to ignore all the "official" notices and
announcements that we post in our group and just see the #other stuff.

Again, they can, but only in a context that includes messages from all their group subscriptions.

A solution to this would be to have new checkboxes on the Filter (Advanced Search) panel which allows restricting the Topics list to those which have (or have not) been Muted or Followed. Yes, I'm still banging my drum for this feature.

You click on the "Hashtags" category; then click on a particular
hashtag. That will filter your view to only those message posts having
the selected hashtag. In this way you can view all the various
categories of hashtagged message posts.
...
The solution is to require that *all* message posts have a hashtag
(the "Hashtags Required" option in the group settings).

It seems to me that a far simpler and more direct solution would be to have something on the Hashtags page that you can click to get a list of un-tagged messages.

Shal