If you extend the logic to other sentences they become similarly problematic. In other words, clearly it's bad to execute an innocent person. But it's also bad to imprison them for life, or 30 years or 20 years or 10 years and so on. There's no clear reason to draw the line at execution but not at life imprisonment.
You can let someone go if you found out they were innocent. This is still bad that they serviced time in prison while innocent.
However it is currently impossible to unkill someone, which makes executing someone for a crime they didn't commit significantly worse, this seems a reasonable place to draw a line to me.
If you extend the logic to other sentences they become similarly problematic. In other words, clearly it's bad to execute an innocent person. But it's also bad to imprison them for life, or 30 years or 20 years or 10 years and so on. There's no clear reason to draw the line at execution but not at life imprisonment.