For example, if we put our minds and money to it would it be possible completely wipe out invasive species like Old World starlings or English sparrows as specific species and not harm other native United States birds?
Both species you mentioned look similar to or flock with native birds so it would probably be infeasible to eradicate them by direct lethal means (e.g., bounties) without causing harm to native species.
Even if we could develop a specific avicide, virus, or sterilizing agent, it would be nearly impossible to distribute/dose to completely eliminate the species as the range is so extensive.
And the reason they are successful invaders is because they fill an ecological niche better than the native species. So the answer to your question is no.
The Deep South has a number of seriously invasive species including Kudzu, nutria and fire ants. All of them are so extremely successful in their new environment that includes few predators for them that eradication is impossible unless some targeted biological agents like viruses are discovered. Kudzu can grow as fast as 2 feet a day during the summer destroying entire properties. Fire ants can form mounds that are greater than an acre. Nutria threaten the entire New Orleans levy system and regular landowners cannot keep up with them at all s professional snipers kill them night after night. Invasive species are a series concern and it is almost impossible to get rid of those. Keeping them at bay is the best anyone can hop for.
Be careful what you wish for.
You’ve been warned!
In most cases, probably no. Once they get entrenched, the best you can hope for is some sort of control. It may be feasible in some cases to introduce a control or other biological control agent, but this won’t normally eradicate the species.
Shagnasty listed ones I was thinking of, excepting the formosan termite and africanized honeybee.