Here's my generic death penalty rhetoric:
The state should have the power to kill people who commit serious crimes. However, the system by which that power is administered should be made as fair as is possible. It must be impossible for an innocent man to slip through the system and wind up with a needle and a last meal, otherwise we must abstain from executions and use life imprisonment.
Would I pay to see an execution, or watch it if it was televised? Probably not. Even if it was bin Laden? Nope.
On the issue of private vs. public executions in general, and your theory that the sterilization of the process has in a way preserved the practice, it's hard to say. Clearly we now have the expectation that an execution will be the capstone in a long process of appeals and hearings designed to make sure that the trial was fair and that the convict has ample opportunity to prove his innocence. The bureaucratic process ends with a bureaucratic execution.
"Electricity will be passed through your body until you are dead; may God have mercy on your soul, and on the people of the great state of Alabama."
Interestingly, public executions died off as mass media emerged. Even early in the early 20th century public hangings still took place. Part of it is that we demand that executions be as humane as possible. The electric chair was a rather effecient way of killing a man (don't know if any women ever got fried), but it was still brutal. Modern sensibilities demand this humanity in the process. The idea is that some abstract justice is served, not that anyone gets pleasure out of it.
When they did firing squads, they had teams of 4 or so shooters, each with one bullet. They picked the bullets at random, and 3 of the 4 were blanks. They fired simultaneously, and no one knew which shooter fired the kill shot. In some states, lethal injection is done the same way, complete with dummy IVs.
http://people.howstuffworks.com/lethal-injection4.htm
Interestingly, Thailand used to execute by
machine gun.