This is a common cliché that would merit some discussion. 'Healthy' is a problematic concept to apply to a machine and ecosystems are like machines in many senses. A 'healthy ecosystem' in this context is often a synonym of 'convenient for hunters'.
Ecosystems without apex predators are more unstable but not necesarily in danger. A breed colony of emperor penguins or a small lake in winter after the resident couple of loons migrate is not an ecosystem about to collapsing or that must be fixed. Many desertic, arctic and pelagic areas are free of apex predators 340 days/year. Apex predators quit and recolonize areas constantly. Is a natural process.
If preys can fly/run away and are free to quit the area the ecosystem will be fine.
If not, a smaller predator will be quickly 'hired' for the job. Could be a desert fox, a parasite or a bacteria, but the ecosystem is constantly fixing itself and will be quickly rebuild with the remaining items.
Hunting can be a temporary solution, but is neither the only answer to this problem nor the best necessarily (and is an obstacle to the recolonization of the area for apex predators in the future, that should be the real goal here). You can always explore other valid options: relocating the surplus animals, creating ecologic corridors, etc, etc...