"Genome" is used so loosely here (it always is in Science/Nature + Phylogeny papers), it's not the full totality of RNA/DNA in the organism the that is being compared. Similar sequences are aligned into "genes", with much data being excluded along the way. The repeatability of the overal analysis from start to finish is notoriously impossible, as there are so many parameters at different stages to keep track of, including versions of software used, etc. There is little to now appreciation/understanding of what CI, virtualization, etc. might add to this process yet, but it shoudl come. This isn't to say the methods are fundamentally flawed, it is to say we have long way to grow to truly incrementally build on the results of prior giants in a more robust way.
Long-branches, like the bird in question, happen in pretty much all clades of life from what I've seen. I suspect we will need a richer phylogentic model that takes into account things like epi-genetics, protein-folding, gene-rearangement etc. if we are to reach a higher level of resolution (or understanding) of the far-off-corners of the TOL.
Its a hard problem. My understanding is the phylogenic trees isn't as an exact science as you would think. Its "NP hard" so they use heuristics to get a good estimate.
The point of the article is that the tree you come up with changes depending on which set of genes you look at, meaning that there's no one correct tree: each gene is inherited at least partially independently, meaning that a simple branching tree is the wrong data structure to use.
Long-branches, like the bird in question, happen in pretty much all clades of life from what I've seen. I suspect we will need a richer phylogentic model that takes into account things like epi-genetics, protein-folding, gene-rearangement etc. if we are to reach a higher level of resolution (or understanding) of the far-off-corners of the TOL.