Their lust for power did lead the people who have taken over the Party to do just that--and now the Party, as an organization, is loathe even to criticize their attempt at subverting democracy by force.
That the Democrats' moving to the right did allow the Republican Party to move further to the right is not the same thing as "driving" that Party there. You made the point yourself that they could have opted to say in the center and the nation would have had a massive center-right political force shared by the two parties. The greatest irony is how little difference in actual policies and programs that would have meant, as opposed to what government has done with Republicans engaging scorched-earth partisanship.