I honestly don't know that you can force most animals to breed together, unless through artificial insemination. Granted, many of the hybrids that have occurred would not have happened if we didn't keep such diverse flocks, if people understood species (peachface lovebirds and black masked lovebirds are not different colors of lovebirds, they are separate species of lovebirds! No, a black cap/maroon bellied conure is not a type of green cheek conure) and could tell them apart.
*MANY* hybrids are fertile, including large conure x mini macaw hybrids (sun/jenday/hahns hybrids), so it would seem odd to me that different species of cockatoos would result in sterile offspring. I'm sure something such as sulphur crested cockatoo x citron hybrids would be fertile. I imagine the umbrella x muloccan hybrids are also fertile... not sure about galah or major mitchel hybrids.