Cloudmin-Pro Cluster

SYSTEM INFORMATION
OS type and version debian bullseye 11.6 arm64
Webmin version 2.10
Cloudmin version 9.7 GPL

Is it necessary to install webmin on every physical LXC host or is it enough to enable ssh access with a ssh-keyfile?

No, ssh-based features should not require Webmin. But it may be useful to have Webmin for other features.