This could be automated. For example, both Ansible and Fabric allow you to enter your sudo password once (per run), and then it will enter it into the sudo prompt for you when executing remote commands. The same thing could be worked out for this (though I agree that it's a less flexible solution).
It's the same method. It uses something like expect to scrape the output of the commands, and determine that it needs to enter a password. That framework could be extended to do the same thing for git/ssh prompts on the remote machine. These tools don't currently do this, but I was just stating that it would be possible to enter the password once even when running this across multiple machines. Note that I'm not necessarily encouraging that as a solution.
For example, I know that one of them (probably Fabric) defines the sudo prompt (via the -p flag or SUDO_PROMPT environment variable) to be able to pick out a sudo prompt in the text output of the command, so that it knows it needs to respond with the password.
The problem with that is that then to automate it you need the password in cleartext in memory. A core dump, ptrace, or anything like that and you've just given up all access.
