This is a great introductory summary! I would just add that Graeber describes the non-monetary economies in somewhat less idyllic terms.

In societies without money, they generally distribute goods on an egalitarian-like basis. This is done through “gift exchange” and debt.

Some societies used egalitarian/needs-based distribution, e.g. the Iroquois with their Longhouses. But most were based on various personal relationships rather than an overarching moral system. (The gift economies were infamously competitive and could lead to mutual ruination.) So the neighbors lending each other things on trust was more about pursuing private interest under the existing social conditions rather than collective good as such - you would still have the rich and poor. But the result was communities based on trust, interdependence, and traditional autonomy, which was a "primitive", but peaceful and viable way to live. Even when neighbors were deeply indebted to each other, they'd still have to treat each other as equals, take each other's needs into consideration, and in general negotiate terms based on ability to pay rather than "you owe me 6% interest because the paper says so, motherfucker".

Graber's great thesis is that money and debt expressed in terms of money changed everything, because they made rational and coldly mathematical the relationships that had once been warm and human. They helped reduce persons to private property stripped of rights. But this same rationality also gave rise to philosophy, which eventually brought about the radical ideas about the abolition of slavery, money, markets, and individualism...

So there's a great dialectic in the historical vision he paints. Over and over in history money sows the seeds of its own destruction, but is recreated again by state actors seeking control only to blow up in everyone's faces again. It's not simply 'primitive communism good, capitalism bad'.

Again, your summary is great. I just want to rave about how much more awesome there is in Graeber's Debt book.