I don't understand it either. What's more, it seems to me that even people who support the death penalty should be willing to limit its use to only those cases where physical evidence exists. Given the number of cases that have been overturned in recent years, such a limitation should be a no-brainer. But unfortunately it's not.
My experience has been pretty much the same as yours. I used to support it but changed my mind a few years ago. However, it was many years before that that I was convinced that we should only allow the death penalty in cases with strong physical evidence. Oddly enough, it was an evangelical Christian who argued that point on TV and convinced me of it (I think it was Billy Graham, but I'm not 100% certain) about 20 years ago.
One thing I've noticed since changing my mind is that the phrase "in order to provide justice for the family of the victim" sounds an awful lot like revenge.
I agree with you, bobbins, and more than anything else I agree with your Reason 1. In the present case, after 26 years what can it be but revenge? Revenge is a seductive notion, one you can fall for without realizing it.
Why should justice for a victim be a consideration when deciding whether someone should live or die? A person should be convicted and punished based on the evidence alone, not considerations of justice for victims. It sounds harsh to say it, but "justice" in this sense is pure revenge.