You've given an excellent answer to the question Why is Dropbox's LAN Sync so slow? but the more interesting question is Why don't the Dropbox engineers change how LAN Sync works so it won't be so slow anymore?
No, it doesn't download it from the master server. It transfers directly between the computers on the LAN. The only thing that has to go through the Dropbox servers is the master file list.
I can understand why, particularly given Dropbox's aim of being as simple as it can possibly be. At the moment the rule is simple. At the moment the cloud can be relied on as the authorative copy which everyone syncs with. If I have 3 or more local clients, all talking and sharing over LAN, many issues get very complicated.
It seems like it could be mitigated by having the client hash the file and notify the cloud of the hash of the file it intends to upload first. But I think your explanation for why they don't currently do LAN-only sharing is correct.
It does do that; I don't understand what you think it'll fix when it comes to multiple clients.
Do you want Dropbox to make a ghost entry with the contents and the hash, then start transferring it over LAN right away? That's going to be confusing when a ghosted file doesn't behave the same as a normal unsynced file. Especially if you start doing collaborative work and the version history is broken because that's cloud-based.
Also, uploading the data to Dropbox server, and then downloading again, wastes valuable quota for those under a data cap.