My first guess would be that this might be a case where the page cache still displays the old 'results'. In case you have any kind of cache plugin installed, try to clear the cache on all layers (page, assets, CDN, cloudflare etc..). The fact that the theme is not changing is very likely pointing to a possible cache problem.
There is a built in plugin cache, which is enabled by default. I suggest turning that off for a while, so it does not mix up anything until you are testing: https://i.imgur.com/rseVi9w.png
Multisite installations can be very tricky, because WordPress usually does not allow accessing post types from cross network, where the post type is not available only on certain subsites. So if bbPress is enabled on site A, but not on site B, and you want to display bbPress related content on site B, then it is most likely not going to work, as bbPress is not available there. Most plugins support sitewide activation, that usually solves these issues. As you noted, this probably explains why the content was not displayed on a site where the bbPress plugin was not activated via the network.Best,
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