I don't agree that the banning admin should essentially have a veto over their unban, and I think any admin should be able to handle a ban appeal. I don't think any policy has ever said that the banning admin has the last word on an appeal (unless there is a new policy). I assume we are referring to jwmphall's ban: I think there was little that the banning admin could add especially as there were screenshots. If I were an admin, I would likely have unbanned jwmphall instead of voting (seemed like section 3 to me), and I believe this would be supported by the most recent version of policy that I am aware of.
I think admins should be able to handle appeals against bans from others. This isn't about distrust of admins. Instead, I think it is more efficient to allow any admin to deal with an appeal, and also means a fresh pair of eyes can look over the details of the case.