LXC is daemonless, there is no process hanging around after the container start, so it starts the container and uses any privileges required to setup things like networking, mounts etc and then drops privileges.
LXC had unprivileged container support since 2013 so that part is fairly mature now. 'Unprivileged' in this case means the container process itself is running as a normal user.
LXC does have a container manager though, which is a single process that stays alive for the life of a single container. Within runc (the runtime Docker uses), we don't have a container manager but the downside is that now the upper level needs to keep alive the descriptors and other kernel objects that allow for safe container management by the runtime.
[I maintain runc, and collaborate with the LXC folks.]
LXC had unprivileged container support since 2013 so that part is fairly mature now. 'Unprivileged' in this case means the container process itself is running as a normal user.