I think this is the best thing the Republicans can do. Keep talking about the issue, asking the Democrats for their solutions, running out the clock.
The press will have a hard time painting the Republicans as roadblocking any legislation since they want to “hammer” out the details. And the Democrats will look more and more feckless as they ramp up the vitriol but fail to do anything.
The sad thing is, you are right, they should diffuse the current liberal outcry by acting like they want to do something, let the temperature settle, then end with nothing because the Democrats are “unreasonable.” The problem is, the base is so jaded by their previous failures and outright attacks on their own agenda, there is no patience for this political theater. Either way this ends, Cornyn will not be plauded by conservatives, and Republicans may pay a price.
While offering none of his own. It's just kabuki theater. Neither side has anything to offer that might make a difference.