The purpose of the package setting "Package Servers Automatically with Manual Prestaging" is to help throttle networks so that packages are not being downloaded by package servers unless a client within the package server's site requests information about a package with this setting. For more info on this setting see "HOWTO45722"
In some cases this setting was working reliably and then suddenly stopped working. Package Servers were not being instructed to download packages with this setting and making them ready, even when clients in their site requested information on them.
A core setting "AgentPackageDownloadFromLocalPSOnly" was enabled without understanding what it does.
The purpose of "AgentPackageDownloadFromLocalPSOnly" discussed in TECH236013, is to prevent clients from coming directly to the NS for package downloads after they have been unable to do so (for various reasons) for a specified amount of time (default is 240 minutes).
This setting was turned on which resulted in the package servers being put in a form of "constrained" mode.
Turn this setting off. If you wish to prevent clients from downloading directly from the NS there is another way to do it. Set the core setting "MaxAgentDownloadTryingTimeMins" to 0.