He claims that nuclear power can't do anything because we can't build plants fast enough and that plant-construction costs have been rising.

The first is a red-herring. It depends a lot on climate projections and so forth. And since it's faster to expand nuclear power than any other emissions-free power, it's not actually an argument not to do it.

The second is more problematic. Cost. Maybe we've lost the ability to build cheap nuclear plants like we could in the 1960s. Maybe modern regulatory and labor problems are unsolvable. But I don't think so. They're a heck of a lot easier to solve than basic technology problems, which is what we're still running into with other forms of energy.