First, they're not opposed to a lower corporate rate per se. They are opposed to a corporate tax rate cut that isn't offset by some tax increase elsewhere.
Second, the de facto tax rate is well below the de jure tax rate, due to the giant mess of loopholes and tax incentives we've accumulated. There's support for bringing the rate down and closing enough loopholes to keep the actual tax burden more or less constant, but that doesn't appear to be what's happening.
Third, their view of the European model is of broad based individual taxes coupled with relatively aggressive income redistribution. Their view of the American model is of very top heavy progressively weighted taxes coupled with minimal redistribution. There's a lot of support to moving toward the European model, but only as a holistic transition. European taxes with an American welfare state is seen as the worst of both worlds.