The bolded is the myth. The truth is, the modern republican party has never held a supermajority in the United States. Without a supermajority, the democrats can block and stonewall them.
Take for example the Bush years, 2000-2008. Some people want to say the republicans had total power, but they did not. For one, Bush barely defeated Al Gore in 2000, via Supreme Court decision. He had no mandate to govern. Then
Senator Jim Jeffords bolted from the GOP 4 months after Bush took office, which handed the Senate to the democrats. The democrats controlled the Senate in Bush's first term. Bush couldn't make major changes domestically in that environment.
Bush won re-election in 2004, and his party held control of both houses on congress. But they didn't have a supermajority. They couldn't push their domestic agenda without getting stonewalled by democrats. Even then, Bush opened his second term with an attempt to
privatize social security , which is a MAJOR tentpole of fiscal conservatism. But they didn't have enough power in Washington to overcome democrat opposition, and it failed.
When Obama won in 2008, he won in a landslide and his party obtained a supermajority in congress. That allowed Obama to ram through Obamacare on a straight party vote.
The myth is that republicans aren't fiscally conservative. But as we have seen with Bush, they tried to privatize social security in 2005. They just didn't have enough power. What we need is for the republicans to control the white house plus a supermajority in congress. Given their past history, they should use it to enact a fiscally-conservative agenda.