For the record, giving extra votes to people with children is not a lunatic idea, in my opinion at least. Matty Yglesias has also toyed around with the idea. I would prefer to lower the voting age to 12 or 14 or something like that, but the principle is similar. Don't get me wrong -- I don't wholeheartedly endorse the idea, but there is some merit to it if you strip away the insults and culture-war BS that Vance brings to it.
Basically, the idea is that it's pretty hard to convince old people to support long-term investments. Here I am talking about a) physical infrastructure like a better electricity grid, federally funded research for new technologies, b) education, both in terms of breadth and depth, and c) long-term fiscal solvency. It's not hard to see why -- they won't be around to enjoy them! It's not too different in kind from the problem often facing publicly traded companies -- that the investors, who are often around for only the short- or medium-term, want to slash R&D to juice current earnings, or who judge managers by their ability to hit earnings estimates multiple quarters in a row over their ability to build strong organizations for long-term growth.
Giving additional voice to the people whose interests are more aligned with the long-term health of the country is likely to promote, over time, better management for long-term prosperity. That is the idea, in a nutshell.
The downsides of the plan are evident to all and do not require elaboration. I am personally agnostic about the idea, as I see the advantages and disadvantages. Maybe this is the corporate law professor in me talking, but bad things tend to happen to institutions over time when you give power to people who don't really care about the success of those institutions. It's the main reason why equity grants became such an important part of executive compensation.