I doubt any Republican who uses that argument would want the US to actually become one of those "we are a Republic, not a democracy" countries like the People's Republic of China, the Islamic Republic of Iran or the Bolivarian Republic of Venezuela for example 
All 3 of those are just as much of a Republic as the US. It's just that they are not democratic republics.
Becoming a non-democratic Republic with a dominant or even single-party system actually does sound a lot like what those Republicans want
