one thing that would be great would be if it was setup in such a way that the API gateway / lambda functions had really simple, file based mappings on a virtual server that then automagically took care of deployment / scaling for you.
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.