Andya,
I don't think you're off base here. As you said, determining the ecological impact of a given species is very difficult. However, a trophic cascade leading to severe ecosystem disruption usually begins at either the bottom or the top of the energy web in a community. Birds normally don't fall into this category. Extinction of avian populations CAN have secondary effects (I'm thinking of the extinction of the lowland populations of Hawaiian honeycreepers via avian malaria leading to the extinction of five endemic species of
Hibiscadelphus - in this case the plants and birds had a tight mutualistic relationship). In general, there are enough other members of the particular species' guilds to "pick up the slack" when one or another population disappears.
Birds are more often the secondary or tertiary victims of the elimination of one end or the other. One example is the extinction of numerous avian species populations on Barro Colorado Island (Lake Gatun, Panama) as a tertiary effect of the extinction of the local populations of
Puma concolor and
Panthera onca. The elimination of these top predators led to a population explosion in ground foraging "mid-level" mammals like the collared peccary (
Tayassu tajaca) and coatimundis (
Nasua narica). The explosion was devastating for ground nesting birds like the great currassow (
Cax rubra), the marbled wood-quail (
Odontophorus gujanensis), the rufous vented ground cuckoo (
Neomorphus geoffroyi), and the black faced antthrush (
Formicarius analis), all of which are now extinct on the island.
IOW, you're right - the extinction of Indonesia's endemic rhinos will likely have a larger and more profound effect on the ecosystem than the elimination of a few bird species.