"serious crime exists in jurisdictions that have capital punishment => it doesn't prevent serious crime"
You're putting the cart before the horse: NOTHING prevents serious crime but capital punishment certainly prevents repeat offending; moreover, what persuades you that the rate of serious offending wouldn't be even higher in the absence of the "ultimate deterrent"?
"Life sentences for serious crimes mean zero chance of reoffences => no need for capital punishment in order to prevent reoffence"
Dream on! In 2022-2023, there were 35,586 criminal offences committed in UK prisons; does that sound as though prisons are preventing reoffending to you?
"the severity of punishment doesn't really seem to have a high degree of correlation with deterrent. It's more about the chance of being caught.."
Rubbish! The problem is that being caught is just a joke, and there is NO "severity" of punishment; a combination of being caught and suffering serious punishment is what would really deter people from offending in the first place, let alone reoffending.
Don't get me wrong: in principle, I am against capital punishment; the idea that the state has the power to decide to end someone's life is definitely something that I have great qualms about. However, that is already the case, whether we like it or not. The police have carried out public executions of major criminals, such as people guilty of being Brazilian electricians in a public place, barristers having a nervous breakdown in their own home, "Irishmen" carrying loaded table legs home et al. Did any of those stop the practice of allowing gung-ho coppers to run loose on our streets with loaded firearms?