It certainly does seem a throwback to old managed PHP environments. No need to set anything up, just provide your PHP code over FTP and off you go!
cp -r / symlinks could take care of the whole staging / lambda version aliasing stuff.
deployment is just scp'ing a zip file to a certain defined location.
all that insane complexity of configuring API gateway is replaced with some config.json or js file export defining what lambda to call and what permissions to use.
then to back the whole thing up you could just rsync it back to your local machine ... hmm though realistically you'd probably want to invert things and develop this whole hierarchy in a git repo that could be checkouted on that virtual server, or it could just natively understand git and branches just become the stages / alias as soon as you push them so no need to ssh in and pull.