” Will it continue to back a conservative, narrow view of government (Tea Party), or will it shift leftward and get more mainstream? “
I have to quibble with the question’s implicit premise. As you’ve already stated, the Left has shifted to the center due to the leadership of President Clinton. That is true. Universal idealism has been left to the far-left and I don’t expect to see it emerge as a force anytime soon. A corresponding shift occurred on the Right when many conservatives shifted leftward, in effect, leaving the neocons, theoconservative-fundamentalists, libertarians and extremely low-information reactionaries in charge of the Party. That is the state of affairs today.
None of those in the present-day GOP espouse mainstream conservative philosophy with the exception of the smattering of Republicans who hold strong positions on fiscal responsibility indistinguishable from centrists and not unlike the majority of liberals. And far too many liberals don’t know this about conservatives because of their strong impulse to use “conservative” for a catch-all pejorative in the same way Republicans use “liberal” as a catch-all for any negative criticism of anyone who opposes their view. So it is correct to say Republicans hold a “narrow” view of government rather that the conservative “limited” view of government of classic conservatism. “Narrow” implying “narrow mined view”, is what remains of the majority of the GOP and it is the opposite of conservatism. Conservatives are intellectually flexible. I know liberals like to stake ownership of intellectual flexibility and it would be hard to make a case against that claim from a conservative perspective. But it’s not a trait exclusive to liberals, nor is it a given that liberals are flexible when they may actually be simply exercising limitless compassion (sounds like a rock band: “Compassion Unlimited”) on an issue that merits deeper thought.