Do I need `mdadm --monitor`?

Yes.
It is recommended to use it if you want to automatically track the state of your RAID arrays and receive timely notifications about failures or degradation.
It is launched either as a daemon (through a systemd service, a script in /etc/init.d/ or similar), or periodically through cron.
If you do not configure this mechanism, then when one of the disks fails (or in the event of other RAID problems), you will not receive an immediate notification.

Typically, for production servers (or systems where it is important to learn about a failure quickly), RAID monitoring is highly desirable.
Sometimes the failure of one disk is unnoticed by the user until a second one fails.
As a result, the entire array can end up in a disastrous condition.
Monitoring allows you to promptly replace the disk and prevent more serious consequences.