I'm not knowledgeable about paleolithic peoples, but the Mori of NZ and the pan-Pacific peoples seemed to have warrior classes. I think they are stone age.
The Mauri and various other pacific islander peoples were/are agriculturalists. I don't know what level of technology they had when the Europeans arrived. They did have a warrior culture. The difference between being a hunter-gatherer and an agriculturalist is much more relevant with regards to whether you have a warrior culture or not, than the presence/absence of metal (though metal gives you a big advantage in making war weapons). As an agriculturalist, you have land, crops and a settlement to defend. If no-one started using metal, it doesn't mean that the other developments we associate with the bronze age (larger city states, armies, warriors, development of other trades) won't have happened.
Compare with Native Australians and Tazmanians who were hunter-gatherers and had no concept of warfare in their culture at all* and their only weapons were designed for hunting animals. They were subjected to genocide by the European colonialists which they had no way to fight back from, which is so beyond horrific I can't even begin to find words to describe it.
*interpersonal violence existed of course, but while two people can fight each other with weapons designed for hunting kangaroos, they can't fight off a whole army that had guns
NZ, Fiji, Tonga and Samoa are great rugby nations. They take "beat the colonials at their own game" very seriously.
(NZ has a mix of Mauri and European-descended players and a strong Mauri influence including the Haka which Mauri players brought into the game more than 100 years ago, afaik.) No, I still haven't got over England women being beaten by NZ in the world cup last month.
(Kudos to NZ who played amazingly though, they deserved to win, England played well but NZ played better.) Fiji men's 7s team won the Olympics - also beating England in the final.
While NZ have the Haka, the other pacific island teams have their own version of the Haka, i.e. from their own cultures which they have also adopted into rugby.
Admittedly, I know more about pacific island cultures from rugby than anything else...
I think economics needs to be considered. If you hunting and gathering, a lot of time is spent doing that, hence less time for warring. Note: time = food.
Definitely economics makes a huge difference. Agriculture caused some massive shifts in human society. The need to defend land/crops and settlements leading to warriors and war is just one thing. The ability to accumulate wealth led to more inequality in society (rich/poor, social class etc), and also sexism arose in agricultural societies where women's ability to contribute economically was limited by pregnancy and looking after small children. (In most hunter-gatherer societies, women obtain a significant proportion of the food by gathering, while pregnant and with small children in tow. However, you can't be a warrior or do heavy work on the farm while pregnant or looking after small kids, so where there's less ability for women to contribute economically, their status in society falls.)
Also, agriculture is a more precarious way of life... if your crop fails you could be looking at a long period of famine. Whereas if the hunt fails a hunter-gatherer only goes hungry for a day or so, and if the animals move way, they go follow them. There are interesting differences in religion/spiritual beliefs between hunter-gatherers and agriculturalists, with more rigid superstition in the latter. Human sacrifice is probably what comes to mind (hunter-gatherers have no motivation whatsoever to do anything like that, but agriculturalists who feel at the mercy of harsh gods who punish them by making their crops die do) but also some of the very rigid rules/beliefs in modern religion date back to the neolithic and bronze age.
I'm not saying all agriculturalists were harsh, superstitious, male chauvinists. The degree to which these things exist in various societies (all, not just agriculturalists) does seem to depend on economic factors. More precarious/higher likelihood of crops failing/having no food = harsher, more rigidly superstitious religion. Less ability for women to contribute economically = lower status of women. Ability to accumulate wealth = rich/poor/social class inequality, etc.
All the above information is looking at general trends. There is massive variation between human societies and there are many other factors besides the above that affect different societies' behaviour and attitudes. You can usually find economic reasons why things are the way they are in any society (including our own) and why things change over time (including the shift from agricultural to industrial).
I met tribes in Borneo that practiced slash and burn agriculture and did hunting and gathering. But the upland rice they grew provided a lot of calories to spend rime doing other things. They were former headhunters, which is not the same thing as warfare. Collecting a head was gaining the 'power' of that person. An analogy would be getting a high school athletic letter, I think: you gain prestige.
I don't know what a high school athletic letter is. It's not something that happens in Britain. Is it like getting a blue at Oxford or Cambridge?
Yeah having discussed the downside of agriculture, the upside is it leads to larger populations (it gives the ability to sustain a larger population on less land) and not needing to have so many people spending so much time just getting food, which then leads to the emergence of trades and people finding different ways to make a living, specialising in what they're good at (making stuff, specialising in trades, etc) rather than everyone spending time hunting and/or gathering and making things on the side. There are massive advantages to agriculture otherwise it wouldn't have been taken on so widely.
ETA: I don't have a huge amount of knowledge about various modern peoples (modern for me is anything in the last 35,000 years or so) albeit that I did learn about general trends like those listed above at uni, plus studying a few modern hunter-gatherer societies but the main focus was on human evolution and the palaeolithic era. There has to be some caution with regards to how much similarity there is between modern hunter-gatherers and ancient hunter-gatherers because change and development in society is ongoing and just because a culture never bothered with agriculture (or never used metals or whatever) doesn't mean they didn't continue to develop in other ways. Also going back on topic, there are modern hunter-gatherers that use metal, albeit trading for it rather than smelting it themselves. People are highly versatile and adapt their way of life to what's needed and what's available. So even if you have a fictional neolithic city that comes across metal through whatever means they do in the OP's story, while it's implausible that they could reverse engineer it and it's likely they'd consider it to be magical in some way, they almost certainly would find a use for it, so it's value would be far more than just being considered magical. If it's useful and they can't make it, then it's going to be extremely valuable.