A bit of history. The first universal packaging format was snap by Canonical and used to be called Click apps and it was made for the Ubuntu mobile OS and later to the Ubuntu desktop. Red Hat in response to that created the FlatPak format. The AppImages are community effort. As you can see since both snap and FlatPak are developed and supported by a company they are more widely available and easier to search, install and update them. There are multiple tools for AppImages as well, which can search, install an update, however they are not pre installed or can be installed from the repo on most distro. There are dielstros which ship AppImage support by default with App Store for example Nitrux. You can use AppMan or bauh for managing AppImages. The AppMan has command line interface and bauh is a graphical application. Bauh can also manage snap and FlatPak.
I am using AppMan as it does not require root and it does install the files into my home directory. It uses query parameter instead of search, but the install, update and remove are similar to the apt commands for example. I use AppImages when there is no package in the repository (or only older version) and it is not available as a Flatpak.
Have look at nwg github.com/nwg-piotr/nwg-shellI believe it has what you are looking for. Panel, app drawer, dock, settings. It is a shell for sway and Hyperland.