Scientist here (you can tell by my hat, and the fact that something like 90% of my comments on /. start with "I'm a biologist"). First, the DNA we get is from better preserved remains, which kicks the half life back further (It's in TFA, but not mentioned in the summary). There's still a 'deadline' around 7 MYA, where (allegedly) all the bonds would have pretty much been broken at that point - Frozen remains supposedly have a halflife around 158 kya. It's that dang phosphate backbone that's too willing to run off and go have reactions with any trallop of a molecule that wanders on by.
This means even in the relatively recent past, the amount of DNA we're looking at is pretty dang tiny. Part of the reason ancient DNA is so dang tricky is because the much of what you sequence is not actually what you're interested in - doubly so when you're sequencing something closely related to humans. For example, did some spot sequencing of ancient/historic polar bear remains, and had to toss out a chunk of the data we got back, as it was soil bacteria(/fungi/pollen) contamination. How do we know which is which? We had good scaffolds to align our bear sequences back up again, though not everyone is as fortunate as us.
In addition to being rare, what is left is fairly short. You can imagine if you start putting breaks in at random, your average length is going to start declining rapidly, and then level out at some small value that takes quite a while to get smaller. It'll get there, and given geologic time scales, a lot of what we want is that far back, but it'll take a while.
Finally, what isn't mentioned in this summary is that there was massive variance in the estimates of half-life. Supposedly only 40% of the variance in halflife was explained by age. Preservation, inter-lab differences, and good old fashioned luck probably contribute considerably to variance in half-life.
There are other factors too, but they're boring, and I should probably get work done instead of dragging out this reply.
(And to answer your latter question, Neanderthals have been sequenced whole genome, not just mtDNA).