For servers there’s Docker/Kubernetes/Podman, which is well-established and serves a similar purpose as Flatpak on the desktop

Originally, containerization was for security sandboxing, so if one service gets exploited, the attacker would only have access to the underprivileged container runtime, not the wider system, and simplifying deployment was a nice side-effect. I have serious concerns that the shift to containers means security updates will be applied less frequently because it needs to be done for a lot more services.

For example, the app I work on gets delayed security updates because we have to make similar changes for each of our microservices, which is a fair amount of effort and a relatively low-priority task. If we had a cluster of similarly-configured servers, it would be as simple as updating system libraries, but since everything is wrapped in a container, those each need to be rebuilt and redeployed. As it stands, our SW stack has a number of security advisories flagged by our container hosting service (none seem realistically exploitable), but they are still largely being ignored because of the amount of effort required to keep them updated.

Flatpak/containers are very similar to the old discussion around static vs dynamic linking, except there’s a level of containerization to mitigate exploits escaping the sandbox.

Once I reinstall my home server, all its server software will be containerised

Same. However, that’s because I care a bit more about ease of (re)deployment and less about exploits, because my home server isn’t particularly critical, and certainly not a big target for attackers. I am more likely to migrate to new hardware than to need to pass a security audit.