Here we enter the realm of pure nebulosity, where the indexes and "real" figures stop being vague heuristics where even the order of magnitude is just barely meaningful, and acquire a metaphysical existence of their own, as "real" variables to be calculated to multiple digits of precision, fed into complex mathematical models and policy guidelines, and used to measure reified true, objective value.

I happen to agree that macroeconomists are overdoing it on the level of precision they can provide. Arnold Kilng (himself a macroeconomist) made this same point in a blog post last year: http://econlog.econlib.org/archives/2009/03/paragraphs_to_p.html

So, here is a straightforward question then: how do we know that it is meaningful to do comparison across, say, between the U.S. in 2010 and the U.S. in 1960 or 1910? What argument supports the assumption that the differences between them are small enough?

I would be careful about using a price index over that kind of time frame, I don't actually know how macroeconomists treat it, but I have read books that point out the inherent difficulty of making comparisons over long time periods (where long means more than about 20 years), and that if you're trying to capture differences in standard of living over a long period one should try to account for differences in product quality and product mix over time. Of course that's incredibly hard to do, and I don't know how seriously this issue is treated in macroeconomics, but it should be taken seriously.

Sometimes it's safer to just leave things alone if you don't know what you're doing.

I strongly agree. However, there are two limiting factors when applying to this logic to policy advice: 1) If you don't give a politician any advice, their reaction won't be to do nothing, it will be to do whatever they think is a good idea. The average macroeconomist may not know a lot, but they know enough that their advice will probably help a little. I do think that macroeconomists should be less willing to offer active advice, as opposed to "we don't understand this problem, the best thing to do here is nothing", but politicians have a strong aversion to doing nothing in the face of a crisis, and if their advisers keep telling them to do politically unpalatable things, they'll find advisers that will tell them what to hear.

2) You can't run experiments in macroeconomics, the only way to acquire data on how well an intervention works is to try it (multiple times in multiple countries) and find out how it goes, and even then you end up arguing what would have happened if you did nothing. That means that if you don't try to fix and/or prevent macroeconomic problems you don't get any better information on how to fix future ones. Maybe that's an acceptable trade off, but I'm sure you can see why macroeconomists don't think so. Also bear in mind that what brought macro into its own as a discipline was the Great Depression. Maybe it's worth risking some bumps in the road to try to work out how to stop something like that happening again.

Prominent macroeconomists will in fact agree with me, it's just that they'll claim that their professional rivals are the dilettantes, and only they are true experts who should be listened to.

Yes, it's depressing how much a macroeconomists' opinion on what caused the recent troubles matches up with their political ideology. But it's a function of the low quality of evidence available, in Bayesian terms when you only have access to weak evidence, your prior matters more than when the evidence is strong. The inevitable influence politics has on the discipline doesn't help either. Politicians are all too keen to build up economists who are telling them to do things they want to do anyway.