The simplest solution is probably something like linuxdeployqt, shipped in a self-contained tarball.
A fairly elaborate solution would be to get into the distribution game yourself, via flatpak (please do not use snap, it's Ubuntu lockin nonsense, the server is proprietary, and not every distro ships it at all -- notably, Arch does not do so). Yes, flatpak and snap both constitute essentially an additional distribution layered on top of another one.
AppImage is basically just a fancy tarball with point and click support. It's a packaging tool, not a build tool, which is why it doesn't come with much in the way of system requirements.
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