From the article:
Experimental design
We used zebra finches (Taeniopygia guttata) from the City College breeding colony. Colony management and isolation procedures have been described previously29. Except for the colony experiment, all birds were kept either singly (isolates) or in pairs (one-to-one tutored) in sound attenuation chambers (Supplementary Information, section 3e) betweens days 30 and 120 post hatch. WT songs (n = 52) were obtained from birds raised in two well-established colonies. Isolates (n = 17) were raised by their mothers from day 7 to day 29 post hatch and were kept in complete isolation from day 30 until day 120 or later. One-to-one-tutored birds (n = 13 and 8 for first and later generations, respectively) were randomly selected from 40 breeding pairs and paired with one of six isolate tutors on day 30. For the colony setting, we made a sound isolation chamber from an inoperational 20-ft3 refrigerator (Supplementary Fig. 1). All birds in the colony (except for the three female founders) were the descendants of the founder male.
So they're investigating the learnt and genetic causes of a behavior, and they get their "isolated" group to spend over 3 weeks with their mothers during a rather critical period of development?..
They try to explain this in a section in their Appendix by pointing out that their auditory system isn't well developed until around 7 days of age (so they can't hear anything prior to that) but that's irrelevant to the situation, however, their ultimate proof that this exposure to their mother doesn't affect the development of their song is that they can't sing until after day 25 - so because they can't sing, they obviously can't learn the song!
That's not the best part though, this is:
It is also interesting to note that during the first
days after hatching the parents tend to be very quite (sic), and if the males sing at all, they tend
to sing away from the nest, facing the other way (Tchernichovski et al, Animal Behav.
1998).
So it's okay, because they're usually rather quiet when they sing around the offspring (I guess so they don't wake them, everyone hates a crying baby), and usually they'll face the other way when they do sing (presumably to prevent the offspring from lip reading?).
Since the initial isolated bird had exposure to the song, he obviously learnt the basics and through exposure with other birds, it gradually gets refined to its more efficient form.
Even if we assume that the methodology was sound, the strongest conclusion we can reach from this is that their bird song is constrained by biology in some way (i.e. some sounds are easier to form that others which dictates the basic shape of the song), and over time the birds are able to figure out the best way to put these sounds together.
There is no need to even discuss a possible genetic basis for this.