I disagree, I have the setting in my .ssh/config and the only thing I need to type on A is ssh C and the session via proxy B is established with no need to type anything on B or C.

This is very helpful when I work with cssh as I have it installed on A (my desktop) and then open multiple session via proxy and do not need to type passwords.

And yes the whole Team is doing this – Both of us! No need for a dedicated appliance (or server) for that many people.