Flathub repo misleadingly considered present on Fedora 35 with filtered Flathub
Downloading the Flathub repository file and opening it in Software does not let the user install the repo. There is no install button or obvious way to enable the repository.
This issue persists after killing Software and rebooting.
I think the issue here is enabling third party repos in Fedora 35 means a filtered version of the Flathub repo is already installed. So there is nothing to add from Software's perspective.
The problem is enabling the Flathub repo on Fedora 35 is supposed to disable Fedora's filtered Flathub repo, and enable full Flathub. See https://fedoraproject.org/wiki/Changes/Filtered_Flathub_Applications
I think the lack of icon and Proprietary label are more AppStream data related so out of scope here.
Tested on Fedora 35 with third party repositories enabled.