So, I'm going to come at this from the paleo-end (because I'm a paleontologist), and I think the what-about-budding issue is particularly critical. With really great fossil records, we estimate species duration just by, well, measuring morpho-species duration in the fossil record. Like this recent piece, on my favorite group, which are far too extinct for us to have any species concept other than the morphological:
Or this piece, about durations of fossil mammals:
But I'm really weary about reconstructing species duration down-tree, at least with current methods. For what its worth, Wagner and Erwin (1995; https://books.google.com/books?id=Mx20KDWnPVEC&lpg=PA119&vq=phylogenetic&dq=wagner%20erwin%20speciation&pg=PA87#v=onepage&q&f=false) tried estimating how morphological 'speciation' in the fossil record is due to budding, bifurcation, etc, and found mainly support for budding.
And this whole budding versus bifurcation, its a big headache, too. One issue I became involved in, is that if you have morphologically-static ancestors with descendant species via budding, you should expect your morphological cladograms are going to be full of polytomies:
Plus, if you have some morphologically unchanging taxon, how do you code that for phylogenetic analysis, like tip-dating with fossils? Do you treat every single find of some morpho-species that has 'persisted' since the Paleozoic (and there are some! plays horror music) as independent OTUs with the same morphological characters? Sounds like a nice way to break the Markov model badly.
For what its worth, I've written some simulation code for dealing with this mess, where species are treated as these persistent units with possible buddings, or branchings or 'anagenesis' between them (because it turns out the inferences we'd make about the fossil record might be pretty different if we think a different pattern of differentiation is involved). You can find it in function
simFossilRecord, in R package
paleotree. Perhaps simulations of such patterns might be useful to your situation?