"in order to be considered a different species, the most important factor was cross-fertility, not mating habits."
What TFA didn't say is that the different song is a factor in mating habits. The two lots of finches don't recognise each other as being the same species. If the Big Bird species survives there'll eventually be sufficient genetic drift to break cross-fertility even if it technically exists at present.
Hybridisation has been recorded as a factor in speciation before, e.g. Spartina anglica.
Deciding whether two things are a separate species are not is a black art. Taxonomists can be regarded as two separate species, lumpers and splitters or as two sub-species of Homo sapiens.