Understand this much - even in some of the more execution-happy parts of America, getting a death sentence is really fucking hard. Outside Texas and maybe a few other states, courts are really averse to giving death sentences and even more averse to carrying them out.
First off, you need a conviction of first-degree murder (by any name; here in Ohio, it’s called aggravated murder, which is basically murder plus additional circumstances) to be eligible at all for capital punishment. Then you need to clear a shit-ton or hurdles. You need every damned juror to vote for death - ONE no-vote and it falls apart. You need to clear at least two rounds of appeals of a death verdict - it can fall apart if there was any problems with the prosecution’s case or if there’s any reason established in law that a person can’t be executed. And it’s this point where people get the most bogged down and why executions take so goddamn long to carry out.
If the prosecution’s case is shipshape AND there are no disqualifying circumstances - mental illness, mental handicap, exonerating evidence, a witness recents or the victim’s family asks for life without parole for some reason - then you have to make sure the execution itself is in order. You need the right drugs, the right people on hand, and the governor can’t commute the sentence.
(For purposes of this statement, I’m assuming it’s a state crime; federal crimes such as treason and terrorism can be included at that level, and the President can be the one to commute a sentence.)
Finally there has to be a will to execute the condemned. If a condemned prisoner is rotting away in death row, a lot of justice systems will ask, why bother executing them? Just let them languish on death row, they’ll kick off sooner or later, and the state can spare itself some culpability, publicity and a few bucks in expenses.
Add to that the overwhelming cloud of suspicion that we have over the process itself. Enough people argue, “well, just execute them after they’re convicted and save some money,” often allowing for a quick appeal to a court of last resort. These are probably the same people who would support death as a punishment for anything from murder to rape to harming a child to any violent felony to (insert light offense here) to talking smack about this leader or that celebrity to driving slow in the left lane (author’s note: the last one absolutely should be a felony, but death is too good for those people.)
The problem? It’s a recipe for executing a fuckton of innocent people. And it’s the only way a system like the one described in the OP could even hope to work, with the sheer number of executions that would have to take place. (And forget the one where people get death for killing animals - this isn’t ancient Egypt, where cats were fucking worshipped, although I guess it is plausible to allow people to use lethal force to defend pets if we have precedents dating back to horses and livestock that happen to include working dogs and mousing cats.)
I don’t see a single country that is not an out-and-out authoritarian and/or totalitarian nightmare being OK with executing innocent people. Even the most hardened death penalty supporters in America would rather let a guilty person go free than chance executing an innocent person (well, most of them would, and those who say otherwise will usually wisely shut the fuck up about it.)
So if you want a PoD that allows for mass execution for a metric shit-ton of crimes, you need something that turns America - and probably much of the West - into what amounts to authoritarian democracies. You would need a culture that is OK with executing innocents just to get the guilty ones (picture a hardline fundamentalist church at the forefront of society that preaches, “kill them all and let God save the innocent.”) And in America’s history, I’m aware of no such PoD that could send the country in this direction.