> anyone on the path knows the amount of the payment
Thanks to the commonality of multipath payments, nodes along the path only know a lower bound of the amount, not the definite amount
> it is possible to identify recipients, and senders with high accuracy, by only controlling some of the nodes on the path
I do not think this is true. Let's suppose you control 5 nodes along the path (Charlie, Dave, Edna, Filbert, and Genna) and you start to trace a payment that flowed through your nodes. You trace it back to Bob on the "sender" side and Harry on the "recipient" side. But you can't tell whether Bob is *really* the sender or just another routing node. And you can't tell if Harry is *really* the recipient or just another routing node. All you know is, those nodes were as far as you could trace it.