Unexpected limitations on "Updates" folder?
On one system, before you install the update - enable caching in the sharing preference pane. Save the cache files to an external drive - speed there isn’t much of a limiting factor since the downloads stream well and any recent hard drive or SSD will be almost as fast as the network can serve files.

Then the first machine to do the download will get the file cached. All the rest of the machines will pull from the cached version.

This is much better than trying to stage the updates on each system as the permissions and code checks are designed to not let you easily place a file on macOS for updating.