I'm not sure that public execution is a good thing either as a deterrent or a revenge. I wouldn't want to abase myself by being so bloodthirsty.
Nowadays, we have the technical means to determine if a person is the culprit, mostly by the use of DNA analysis.
The last person to be executed by guillotine in France was convicted of raping and killing a little girl. But even today, with all our technical arsenal, no one knows for certain if he did it. That's what prompted the fight over the death penalty, and finally the ban.
Therefore, before starting to hand death sentences again, I think, IMHO, that we need to define which kinds of crime make the death penalty come into play and what are the rules for that (DNA present, nor present, witness reliability, etc) and alternate sentences.
Then, when it is done, we will have to enshrine it in the Constitution in a way that no politician, can change it, no matter what circumstances can be. Otherwise, we could find ourselves back in the days where stealing bread because we are starving could bring a swift death.
This is my own opinion, one I had more than 20 years to forge since I became an responsible adult.(Majority in France is 18 - I'm not sure about the responsible part
).
I changed from no death penalty to kill them all and now to a selective attitude.