"It's some time I read Hennig's book, but isn't uniqueness a necessary property of a synapomorphy?"
Few people these days have read Hennig's book, for the same reason that few have read any of Darwin's: they're only interesting as part of the history of science. Except for a few quotes found elsewhere, I haven't read them myself.
It is easily possible that Hennig defined synapomorphies by uniqueness. After all, he believed that homoplasy does not exist – that if you get any in your tree, you need to redo your entire matrix from scratch, because you've mistaken something merely similar as fully identical. In the molecular realm, we now know that this is complete nonsense, and it follows that the same holds for morphology.
Personally, I do follow a terminological distinction Hennig made: one clade has autapomorphies (auto- = self), two sister-groups have synapomorphies (syn- = together). But this level of precision is hardly ever needed; I think Hennig had a background of German Idealism and liked to invent terminology for the fun of it.
"what is your argument against summarising the MPT sample using consensus networks?"
I, for one, don't have any; I'm not familiar with consensus networks and need to look into them.
That said, I don't think the fossil record of the taxa sampled in this paper is dense enough to contain any traces of reticulate evolution. If all Paleo- and Eocene birds were sampled, that could be different.
"long-branch attraction issues"
Absolutely, these are real; parsimony is more sensitive to them than model-based methods; and while they're much more common with molecular than with morphological data, they do exist in the latter, too. The best that can be done about this problem is dense sampling of taxa and characters to break up long branches.
"time-slice subsets – Why is this an "arbitrary exclusion"? All molecular trees (those not including ancient DNA samples) represent a time-slice taxon set."
...But that's a bug, not a feature! Whenever molecular phylogeneticists have ancient DNA at their disposal, they use it. They just almost never have any, so they do the best they can and sample the extant taxa.
In principle, it might be interesting to do a phylogenetic analysis of a time slice in the past in order to test how hopeless the task of the molécularistes is. But the fossil data aren't good enough for such a test.
"many palaeontologist reject the notion of ancestors in the fossil record"
This is a misunderstanding.
1) Compared to the diversity of life today, our sample of any other time slice is so small that, statistically, we should expect to find few if any ancestors of any known taxon except in exceptional cases (say large mammals of the Late Pleistocene, or Pliocene midocean diatoms).
2) Outside such exceptional cases, every time a fossil has been proposed to be an ancestor, even if it was explicitly claimed to lack autapomorphies, it was later found to have autapomorphies after all. (Archaeopteryx is the most prominent example.) Obviously, autapomorphies aren't proof that a taxon died without issue, but they do make that hypothesis more parsimonious.
3) If we don't find autapomorphies – which, again, is actually really rare outside the mentioned kinds of exceptional cases –, we can always blame preservation: perhaps all the autapomorphies were in the soft anatomy or just the DNA and aren't preserved.
Zero-length terminal branches are not common in morphological phylogenetic analyses, even though most matrices are made for parsimony and therefore don't code known unique autapomorphies of terminal taxa.
"By limiting your taxon set to same time-slices, you eliminate such time-related topological effects."
But you invite long-branch attraction by refusing to break up the long branches you have.
"If the (let's say) strict consensus tree only has true branches, i.e. reflecting the actual lineage splitting over space and time, then each taxon subset tree would need to show those branches. Or, at least, don't show conflicting alternatives with higher branch support. If it does, the analysis (and the strict consensus tree) has a problem."
The problem is character conflict, which is real. Trees made from subsets of the taxa will not show us which signal is phylogenetic and which are homoplastic; to the contrary, if the phylogenetic signal isn't much stronger than all the noise, they will suffer from accidental sampling bias and end up preferring one of the homoplastic signals over the phylogenetic one just by chance. I've expanded on this in the introduction to my paper cited above.
The solution to noise is to add more noise. In the immortal words of Thomas Holtz, the signal adds up, the noise cancels itself out – if and only if there is enough noise to do statistics with. If your matrix is too small and the phylogenetic signal in it is not unrealistically strong, you run a serious risk of accidental sampling bias.
"Why using a cladogram if you can show a phylogram?"
Probably just out of tradition. I'm not sure if parsimony branch lengths have ever been published in paleontology.
The tradition does have a reason, though: we know how small our matrices are. Add a few characters, and the lengths would be noticeably different. We also know how gappy our matrices are: if we had all the data we lack, almost all branches would be longer, but not all by the same or any other predictable amount. Given that the branch lengths are (generally) more sensitive to these problems than the topology, it makes less sense to publish them.
Model-based approaches try to compensate for all this. They put out branch lengths in expected changes per character, not in actually observed ones. Therefore, when vertebrate paleontologists run Bayesian analyses, we do publish these calculated branch lengths (not only because that's how Bayesian trees are published elsewhere). How well this attempted compensation really works with paleontological datasets has not been tested.