I wouldn't say the general public was very far to the left (if any) on a shopping list of all progressive goals, and even when they are, it's unlikely they are as committed to a particular reform (only up to $50,000 of student debt, by the way) as the rest of the country is to the opposing view / strategy.

But I still agree with your overall assessment of the Republican Party. The trick is to SAY (for the sake of appearance--i.e. as a rhetorical ploy) you're willing to work across the aisle / compromise / achieve consensus between parties, then bait the Republicans to act like Republicans and use their resistance to alienate some of their moderate supporters and then drive home as many reforms as possible.