You actually have two different work flows covered with the examples, software installation and update. The difference is that users likely want to search for an app not knowing how it is called. Google playstore is a perfect example for search where you aren’t restricted to certain categories but get similar programs and all wisdom from the crowd.
On the other hand I doubt Solus can change the world as it was defined years ago. The packages under Linux are tagged in the corset of groups as defined here http://www.tldp.org/HOWTO/RPM-HOWTO/build.html (6.2 Header: Group) and more detailed at https://en.opensuse.org/openSUSE:Package_group_guidelines. Similar is true for Debian packages and I believe the structure is craved in an ISO norm.
What I also miss are some more information. Really well done was the UE for Muon Discover (why not use this in Solus) https://community.kde.org/KDE_Visual_Design_Group/Muon_Discover (as an Arch user I never used this installer nor the Ubuntu software center).
To go more into detail, I always feel confused when the start screen tells me to go to categories (Desktop software, Gaming…) and in parallel to check Third party, Installed or whatever types. It looks like both are related and I get the categories of software for the types from the sidebar. Why not put the updater into a separate tool? Do Solus users really care what app is updated? With the ability to keep the old version. With a lot of hierarchical information. Okay, Uninstall is there but you may get my point.
The button like visualization is not really appealing to me. How about a big friendly icon, with a title, and some sub items showing what actually means. With a proper use of font size and bold it looks nice without any additional control. There are of course many details to flesh out if you go in this direction.
What I’d recommend is to define the requirements with user stories like “John wants to find a game by entering the keyword “chess” and selecting the most appropriate by visual appearance and functionality out of a list of options. He also wants to get similar apps like Go in case of chess.” (John is one of your personas). For the core usability it’s much easier to discuss those user stories than the final designs with shiny images focusing on visual aspects.