It does make little sense not to give the option to all, at least if your non-mage took the arcane knowledge perk. Whilst it is true that First Enchanters in the Circles had to approve the Rite before the Templars could carry it out, when done legally, there are instances where another person in authority could do so. One that springs to mind is when Justinia approves re-tranquilising a mage, who had reversed the procedure when undertaking research on her behalf. Okay, she was reaffirming his original sentence but even so, she was doing it as a non-mage.
I presume the thinking behind this is that only a mage truly appreciates the horror of what you are inflicting. However, that is not sufficient reason, from a lore and role playing perspective, to restrict it to only mages. Perhaps what they should have done for a non-mage is have someone on hand, like Vivienne, to endorse your decision. After all Cullen asks to stand by when you are judging Samson. Of course the only drawback to this idea is that you can decline to recruit Vivienne and of course there is not a mage among your advisors but, even though she isn't a mage, Leliana seems to regard herself as their advocate, so in the absence of Vivienne, Leliana could have been the one to approve a non-mage's sentence of tranquillity on a mage.
My mages really couldn't contemplate doing it, even to Erimond, since that would make them as bad as Corypheus and Tevinter, since Dorian tells us they use it there to silence dissenters and leave them as an example to others. So I tend to opt for imprisoning him, throwing away the key and then leaving him to hear how we have wiped the floor with his god.