Well, to me there's a really obvious reason for not letting a DRG be on stun duty.
Your stun, as all other stuns are too, is on a global cooldown linked with your other skills. For anything that charges quick, it basicly means that you wont be able to do anything other than keeping an eye out for stun, due to it possibly not being available for 2,5 seconds if you just hit an ability, or hit an ability while it's charging even.
So you have a choice, use a DRG for pure damage, or let him just lounge around and just stun whenever it's appropriate. More often than not, there are other jobs out there that are better suited to be on stun duty unfortunately. Jobs that dont contribute as much with damage. Usually a second tank takes that role since they're only there to either jump in when stuff goes horribly bad, or to assist on spawns.
[XI] Surivere of Valefor
[XIV] Sir Surian Bedivere of Behemoth