Yep, yep.
With no facets (and more changes)
I tried out a layout thinking of this and it would look something like:
I took many elements from the advanced search page, without the facets and highlights. Changed a bit the wording of the sorting types and the position of the paging.
How I feel about the absence of facets and highlights: Highlights are not really needed when we are in the context of filtering by tags. Facets, on the other hand, could be very useful if a team tags a lot of pages in different subwikis and stuff.
I made the sections on the right collapsable. These sections are:
- Activity stream for pages tagged with [… ] - renamed Related activity stream to not repeat the tag for no reason. it’s pretty clear that the activity stream is stricly for the pages tagged with the tag he clicked on.
- Related tags - based on @mleduc 's idea, this would showcase other tags used alongside the one chosen by the user. Ideally they would be ordered by hoiw many uses they have (in how many pages they were used alongside the chosen one)
- Search by more tags - based on @mleduc 's idea, this would enable the user to add more tags to the search in an input field. When clicking a search button, he would get redirected to another tag page with all those tags.
With facets
If we want to keep the facets (which I think we should) and make activity stream a secondary usecase, we could do something like this:
I changed the look of the filtering in the right to lose the gray in the background (for a more lightweigt look) and the look of the tag (to allign with the tag look revamp proposal). In the facets, I’ve included the Related tags filter. Activity stream is now at the bottom of the right side and is collapsable.
@CharpentierLucas , @mleduc and everyone else who will stumble upon this post, let me know what you think about all of this when you have the time! 
I will dwelve into how Related tags or Search by more tags would look and work if there is interest for these ideas.