Seeds can float on the sea for years. Shore birds that are used to flying over the open sea do spread across much larger gaps, but the passerines and other birds adapted to life in trees or on the ground generally avoid long flights over open seas and only wind up there if blown by storms - the ocean is a very deadly place for a typical bird with no desirable resources.*

* In the Americas and some parts of Eurasia there's migratory birds which are passerines that fly over the open seas, but they evolved this behavior for adaptive reasons and only fly over the ocean for migration and then when they've fattened themselves up for weeks and prepared for a dangerous over water journey that not all will survive. In their cases body morphology evolved to allow that migratory flight.

I am not an expert, but my understanding is that the line isn't quite so sharp as depicted here. Instead, there's a broad transitional zone (Wallacea) between the Asian and the Australian ecozones, and the Wallace Line and the Lombok Strait is merely the western border of this region, with the eastern border being the actual fault boundary between the Eurasian and the Australian plates (on the Australian side of Timor).

Looking at the geography of the archipelago, the Lombok Strait is the first major gap that would have existed, and it also seems to be the second largest gap in the region, with the larger one existing between Timor and whatever the island is to the northwest of Timor.

What I suspect is going on is that the fauna on Bali is largely adapted to large terrestial landmasses--even during interglacials when the land is disjoint islands rather than contiguous landmasses, Sumatra and Java are large islands (6th largest and 13th largest, in fact), which means there's not a lot of evolutionary pressure for transiting decent straits. By contrast, the region in Lombok and points east is largely somewhat disconnected smaller islands, so their ecological niches are going to be more heavily dominated by fauna that can make the strait crossings more easily. So if you've got some fauna that occasionally make the crossing, they're going to be outcompeted by the locals and don't really last long enough to make viable breeding pairs to set up a population. And Lombok is perhaps as far west as Australian-sourced fauna can make it before their small-islands adaptations start becoming ineffective against the native Sunda fauna.

That is all mostly just me conjecturing, though, I don't have any hard data or sources to back it up.