It's an interesting question and I did a bit of digging, without success. I would expect the best place to find an answer to be in Dumézil (Archaic Roman Religion, 1998), but I can find nothing good. In the section where he discusses signa and portenta (594ff) he steers clear of speculation and reports simply that we know frustratingly little about augury in general. He suspects the practice is old, probably early Iron Age at the latest, and indeed the fact that we find bird-watching in at least one other Italic culture (Umbrian) suggests as much. I went in suspecting a connection to Etruscan, but it seems that was a wrong assumption in this case.

One nitpick: the observance of birds for the Romans was always in pursuit of a single god's sign only, Jupiter Optimus Maximus (see Cicero Leg. 2.20). Dumézil mentions a hierarchy of bird signs, with the activities of lesser birds under a permanent lesser status than others. The king of birds was the eagle, Jupiter's bird, which makes perfect sense, and an eagle's sign would always supercede any other lesser bird's.

Since Dumézil does not speculate, I shall: there is always a pinch of practicality in Greek and Roman ritual practice, and I see it here. The Romans, especially the early Romans, were superstitious to a fault. The practice of consulting Jupiter via bird signs was a pressure relief valve, especially when making big decisions. It prevented Dumézil's paralyzing "psychosis of the sign" and established a hierarchy and "final word" in divine signals. Once the eagle had spoken in her flight and landing upon this or that roof, lesser signs (like a vomiting dog, or a soldier's twisted ankle) could be safely disregarded. Secondly, Jupiter has a connection with birds, namely the eagle, and birds in general were every-day enough to be useful for consultation compared to, say, a meteor shower or a two-headed calf. One could reliably get some kind of sign from birds at any hour of the day. In the Late Republic, the political expediency of augury and the declaration of nefas elements in general was plain, infected with such cynicism and bad faith as to become essentially useless. A good example of Caesar's consulship in 59, during which nearly every business day was declared "unlucky" by his recalcitrant consul colleague Bibulus. The haruspices (examination of sacrificed animal livers) were a useful alternative in such cynical times.