I don't see why wars would be any harder to start without a state. All it would need is the people with the most wealth and control over society to push others in that direction. Just like now.
I wasn't picking sides, just presenting another point of view.
That said, consider the magnitude of wars fought with state backing. I doubt there could have been a private funding or collaboration for the Manhattan project, icbm's, or other tools of global extinction. Only recently have private citizens been successful at launching spacecraft, based on 80+ years of state funded r&d.
Historically large advances in military hardware have required the backing of a state or city state. Standing armies have also historically been either state funded or a form of slavery. There are instances or private armies challenging state powers, of which they became the state themselves. The war of the roses for example.
I think the most central issue these days is fiat funded, unending wars. Eventually a private army would run out of money, this isn't an issue for a state with a central bank, or particularly the world reserve currency.
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