Unsure why you think that copying a console application to a production server is both wrong and unsustainable
.
There are two choices, copy files for the application to the server or create a setup program to install the application.
Not knowing your policies at your workplace, both options are viable outside of who installs or copies the application to the server but either way there is nothing wrong with copying the application to a server.
NuGet packages typically reside in the application folder, see Adding packages to the feed externally and Setting Up A Private Nuget Server (for development)
Note you can point to apps to custom locations
# Set repositoryPath in the user-level config file
nuget config -set repositoryPath=c:\packages
# Set repositoryPath in project-level files
nuget config -set repositoryPath=c:\packages -configfile c:\my.Config
nuget config -set repositoryPath=c:\packages -configfile .\myApp\NuGet.Config
# Set repositoryPath in the computer-level file (requires elevation)
nuget config -set repositoryPath=c:\packages -configfile %ProgramFiles(x86)%\NuGet\Config\NuGet.Config