It's interesting how this debate is morphing into whether or not we should define Republicanism based on some Platonic ideal of what a 'Republican' should be from a historical perspective versus what the majority of people in the party believe today. Personally I favor the latter definition, though I do agree that Badger definitely would have had a strong place in the GOP historically.
"Republican" isn't an ideology, so it's really nothing more than the consensus of what people who label themselves as Republicans believe.