I did some research on this topic. I am also having same doubt now :/
Memory protection is necessary but it's not like OS cannot be implemented without hardware support for address translation.
Though I feel they are talking about the basics. I mean, for a multi-user multi-processing OS, even 2 modes of execution is not necessary. We have privileged mode of execution so that there is fair usage of hardware by every process and to monitor activities so that one process doesn't disturb other process's execution. It doesn't mean that without privileged mode multi-user multi-processing OS cannot be implemented.
Process scheduling, Memory management, File system management are some basic functions of OS. Hence, I think address translation is also chosen as an answer.