Reasonable, but even so, while there are men who do indeed seem to enjoy and seek tribal war-like scenarios (a very common example could be hooliganism, as big chaotic brawls of two opposing groups in the street using no weapon more complex than some blunt implement or rock throwing is probably as close as it comes to warfare in the stone age) they're far from a majority.

Again, anyone wishing to practice violence has a number of sports open as options. I doubt that the lack of a high probability of death is what holds people back, as if these weren't hardcore enough options. In fact most people seek safety in a number of ways, and very few willingly court death or bodily harm. There is no fundamental social problem that would be solved by "reintroduce death games" because:

1. non-death-but-still-pretty-painful games are already a thing and not nearly enough people seek them to suggest a vast unfulfilled need that only needs the right outlet;

2. as things are, if the only change you introduced was to legalize death games, the results would not be significantly different from any other spectacular violent sport, with the probable exception that being a contestant would be more low status than being a boxer or MMA fighter due to the risk of death, and thus, it would not fulfil any broader societal need anyway. It would just open a dangerous loophole for Moloch to inevitably converge towards the creation of a poor people meatgrinder that maximises higher class amusement.

In fact, we already have something of an example of what this would look like - let's indulge another ever-present male urge and Think About the Roman Empire. The Roman Empire is the go-to reference when thinking of death games because gladiatorial battles were basically just that, and in many ways, the Roman Empire was closer to our consumeristic society than many of the societies that came afterwards, so we can actually see the gladiatorial battles as a proper entertainment industry of sorts, which aside from the occasional Emperor-sponsored games had to earn its keep with paying spectators. And what do we observe?

1. the death battles weren't very often to the death. Obviously they were very unsafe by modern standards, but still, training a gladiator to be a good fighter that provided a cool spectacle was expensive and you didn't want to throw that away at the first loss;

2. gladiators may have been very famous and even have fans - but ultimately, they were mostly slaves. Maybe some very high status, pricey slaves, among the most likely to be able to buy their freedom (though given the nature of their job I'd assume not with the best rates of survival-to-manumission), but still slaves. The highest of the lowest remain pretty low. Freemen who chose gladiatorial combat existed, but they were usually considered also very low in the social hierarchy, like prostitutes;

3. while the average male Roman citizen did have to perform military service, I would say the Roman way of war was already different enough from anything you'd have witnessed in the stone age to count as out-of-distribution from the ancestral environment, though of course not as much as anything after the age of gunpowder, let alone modern war. Regardless, most of it wouldn't be exciting man-to-man combat, but gruelling marching, building camps, and/or dying of dysentery in the muck of some Jupiter-forsaken German forest. So I don't think you can count it as satisfying the need already enough. War after all has to always be dictated by maximizing chances of victory, not being fun even by whatever definition one could possibly call deadly combat "fun".

I see no reason why the same exact logic wouldn't apply today. Military service is open to anyone willing but doesn't alone fulfil this supposed need. Economics of gladiatorial death games would quickly converge towards minimizing death anyway, so they'd become a merely slightly deadlier combat sport than MMA or American football. Possibly not even as deadly as some other extreme sports. Being a gladiator would quickly become low status compared to other types of sports, a profession attracting mostly antisocial types and desperados with no other option. So most respectable people with good prospects and social ties, even if they did have such a craving for violence to satisfy, would still not do it because on the balance it wouldn't be important enough to forgo a better paying career, or a family, or a good social status. Because obviously "risking your life only because you crave having a chance of killing others out of sheer bloodlust" is always going to be seen as antisocial and barbaric, even by societies who enjoy watching the resulting death games. The gladiators are merely monkeys dancing for the amusement of their betters.

So the proposal of the OP doesn't solve a problem that we aren't even sure is a problem at all, and introduces for sure more ways for Moloch to exacerbate inequalities and kill people for no good reason.

That looks pretty fun and also fairly distant from a "death battle". I'm down to engage in even more brutal forms of combat, such as with those giant cotton buds while balanced on a beam above an elastic mat.

Well there has to be some compromise here. Can the berserkers get some wild west region where pubs aren't policed as heavily?

i meant two man fistfights, not brawls