No, of course not. Because the notional escapee could as easily escape from Death Row as from Life Imprisonment. Given that the time spent on Death Row is measured in decades, the chances of escape are almost equal.
Now let's ask the same question the other way around. Let's say a guy who murdered someone. But during the trial the threat of the death penalty means that two witnesses don't come forward and the jury narrowly votes to acquit (because if you're killing someone, you want to be darned sure). So now he's free, because locking him up wasn't enough to satisfy your desire for revenge. If he was fated to kill a member of your family, would that enough to change your mind about the death penalty?
If the jury wrongly or perversely acquits him then that's that.
Nothing perverse about it. Juries are asked to judge beyond a reasonable doubt. It's left to them to decide how much doubt is 'reasonable'. Applying the death penalty raises the stakes (which is a reason some people support it, as a deterrent), and thus makes the amount of doubt seen as reasonable much smaller, making reaching a verdict of guilty beyond reasonable doubt a higher standard.
The same goes for witnesses. For murders you're often looking for friends and immediate family members to step forward and say what they saw. You're also asking people who may not know themselves if the guy was guilty or innocent to give the facts that they witnesses. Both groups of witnesses will be much more reluctant to come forward, or to speak up, if their words will kill a guy.
Having the death penalty makes it harder to secure a conviction. Or to put it another way, applying the death penalty means being soft on murder.
But with your logic he could be free to murder again anyway,
Nope, if you can imagine a guy who is fated to escape if locked up, then I can imagine a guy who is fated to be let off if the jury and witnesses know they'll be killing him by finding him guilty. And my scenario is much much more likely than yours.
whereas with mine he wouldn't be, ...
Yes, he would. Because sentencing him to death doesn't remove him from jail for 15-20 years, on average, possibly longer. So the escape rate is about the same.
Look, imagine 100 murderers
1) Your way - death penalty
100 murderers
70 get caught
35 get convicted and sentenced to death
5 escape from Death Row
70 free to kill again. (65 never convicted, 5 escapes)
2) My way - no Death penalty
100 murderers
80 get caught (more witnesses)
60 get convicted (jury effect) and sentenced to life
10 escape from Life Imprisonment
50 Free to kill again (40 never caught, and 10 escapes).
By making it harder to convict murderers, you catch fewer of them, and more people die as a result.
That's what happens when you dilute a worthy goal (stopping murderers from murdering again) with an unworthy goal (wanting to satisfy your desire for vengeance, even if more people die as a result).