I see two issues here, and I'm not sure which one you wanted to focus on. There's the issue of whether or not executions should be made public, and the issue of whether or not a private citizen should choose to view them (or one, or whatever).

Regarding the first issue, I totally agree with what you say. On the other hand, I have an opposing argument that is also compelling to me. You say that if we elect to exercise capital punishment but do not see the results, we may be doing the wrong thing, because it does, after all, make sense that if we are going to support such a grave thing, we had better be capable of grasping the reality of it.

But I think there's another extreme. If all executions were broadcast on national television, for example, and the consequences of our national decision to practice capital punishment were made visible and tangible and apparent to all, eventually I don't see how there could not be a desensitizing to the natural horror we would feel at first. And if we become desensitized to death and executions, what happens from there? Might this be a path that ultimately leads, if unchecked, to a kind of civilization akin to ancient Rome, where gladiators battling to the death were a form of entertainment for the masses? To speculate less dramatically, might we at least become desensitized to executions enough that the basis of support for capital punishment is LESS, not more, grounded in an understanding of its import? Might exposure to executions ultimately make us less cautious about this handling of human life and not more?