Greg Cochran's comment below is worth turning into a post:
There's more to it than that. Tribes often have extremely limited HLA variation, contain only a small subset of the variation that you see in a wider set of Amerindians. Whereas in the old world, even little tiny groups with very low gene flow have lots of different HLA alleles. [Cavalli-Sforza 1994] You'd think that they'd lose those rare alleles by drift, but they don't - has to be frequency-dependent selection, the same force that has kept alleles around for tens of millions of years. But in the Americas, it appears that those frequency-dependent forces simply did not exist. [Slatkin and Muirhead, 2000] So, two things going on, which may or may not modify your conclusions. First, a bottleneck, probably: afterwards, a world in which HLA simply does not matter.
We talked about this subsequent to this comment. Basically in small populations subject to a lot of random genetic drift HLA diversity still remains high because stochastic factors run up against powerful negative frequency dependent selection effects. That is, the rarer the allele, the stronger its fitness advantage. So, as drift drives an allele frequency down it begins to run up against countervailing selective pressures. Just as drift is about to run an allele to extinction the break is slammed and it will "bounce back." This is why HLA variants seem to be almost immortal fragments of the genome. So what happened with Native Americans? Greg's point seems to be that Native American groups were not subject to this particular dynamic where HLA is kept diverse within groups, so convential genetic forces of drift were far more powerful on these loci than in other human groups. What's different? One could posit things like density of population, but the HLA have deep roots well before our own species.