replenishment of wild salmon
The process you propose is called 'stocking', and has been carried out for many different fish species for many years. The practice is, as you suggest, complicated, and more than a little contentious.

In the specific case of Atlantic salmon, Young et al. (2014) provide a non-technical overview of the conservation, population-genetic, and infectious-disease factors at play in stocking decisions. In general, fish raised in hatcheries are often wild-caught, which decreases the wild population, and once they are released, they sometimes have lower survival rates and reproductive success than truly-wild fish. If fish are raised in captivity for multiple generations, they can show evolutionary adaptations for captivity, and once released, captive-adapted fish can introduce harmful genes to the wild population. Similarly, captive-reared fish may be at an increased risk of communicable disease, and may introduce diseases to wild fish after their release.

A further complication from stocking is the impact of the stocked species on the local ecosystem. Many commonly-stocked species are non-native in the areas where they are released, and have negative impacts on the native species in that area. For example, brown trout have been introduced to many waterways in New Zealand, and they are predators of many native species, and compete for food with others. The New Zealand government has trialled an eradication of brown trout from some high-value conservation waterways in order to minimise the ecological damage caused by trout.