outside of bitcoin, how does a network negotiate what the next blockchain should be?
Here's how Bitcoin works.  I suppose we could say that each node can "propose" a different next block.  What this really means is that each node can select a "proposed" next block and start mining to attempt to append that block to the blockchain.  Bitcoin's proof-of-work  basically creates a lottery among all of the nodes who are doing this; it's random which one will win, but there will be only one winner, and once there is a winner, everyone will know which one won.

Nothing stops dishonest nodes from attempting to "propose" many different next blocks and mine from them.  If 100% of the network is dishonest, then you're screwed anyway.  So if you want any security at all, you need to assume that at least 51% of the nodes are honest.  And the crucial part is to look at how the honest nodes will behave.

The way that a protocol works is that it specifies how an honest node should work.  If the node is honest, it will follow the protocol faithfully -- that's more or less the definition of honest, in the world of protocols.  The Bitcoin protocol specifies how an honest node should determine what transactions to include in its proposed next block: include all of the transactions that it is aware of, and that are valid.  It also provides a mechanism to help all of the honest nodes learn of all possible transactions.

So, if at least 51% of the nodes are honest, they'll all "propose" the same next block (since they're all honestly applying the same validity checks) and start mining from that next block.

What about protocols other than Bitcoin?  No one ever promised that other protocols would be secure.  No one ever promised that merely using a blockchain (a hash-chained ledger) is sufficient to ensure security.  Bitcoin is a clever combination of multiple interlocking mechanisms that work together to provide security -- the beauty and magic is in how they fit together.  If you take away some of those mechanisms and keep only others, there's no reason to expect the result to necessarily be secure.  And it's not really possible to analyze the security of an unspecified protocol, where the only information we have about it is that "it uses a blockchain but isn't Bitcoin".