What is the impact of the invasiveness of the European Starling in North America?
First, for the record, I do not know the North American Starling situation well. However, I know of one study (Koenig, 2003) that failed to find strong effects of Starlings on other cavity nesters (i.e. looking at effects from cavity competition). When trying to control for other effects, only one species was potentially negatively effected by Starlings.

Starlings cause quite large economic effects though (Linz et al, 2007), and it wouldn't surprise me if this is the main reason for the control measures. The paper refers to a previous study which estimates that Starlings cause crop damage costs of \$800 million/year as well as \$800 million/year in health costs, due to disease transmission to livestock and humans (both figures from Pimentel et al., 2000).

So my feeling, based on a couple papers and some quick searches, is that the ecological impacts are uncertain (or to a large extent unknown) but that the economic impacts are quite large. Given the North American population sizes of European Starlings it would however surprise me a lot if there weren't any ecological effects on other species (negative and positive). These issues are hard to study though, especially without long-term data of high quality.