WADK Deployment Tools on Windows Server 2008 giving error "The program can't start because api-ms-win-downlevel-advapi32-l4-1-0.dll is missing from your system"

Got this error at an extremely inconvenient time when trying to write some automation scripts for Windows Optional Features using the DISM module as part of the Windows Assessment and Deployment Kit running on Windows Server 2008 R2: "The program can't start because api-ms-win-downlevel-advapi32-l4-1-0.dll is missing from your system"

It turns out that the installer for this particular product isn't reliable when you uninstall and then reinstall the ADK. The problem turned out to be that the PATH variable hadn't been updated on the second install, so the PowerShell module that was invoking the DISM.exe executable wasn't able to find the required DLLs, even though they were in the same path as the executable. Go figure.