All in all I don't know that "payroll-connected API" is any better for you - quite a mouthful and sounds way more techy.
Personally I would like "Payroll verification API" but maybe that's just because we're having this discussion and that fancy will disappear like dust in the wind by the end of the day. It might also preclude you from getting into some of the juicier areas that a16z are talking about (i.e. things that are > just verifiying data), but of course you can always change your tagline :)
FYI this area is of interest as we briefly explored automating some nasty govt systems using selenium or similar (aka RPA). We would run the robot in a VM that was torn down at the end of every session for security. We presented the UI to the user via guacamole - then once they had entered their credentials, the robot took over. Interesting but not the path we took in the end.