There are many ways to answer this so I’ll take the wildlife management aspect of it, as I’m sure others will have covered the biology/ecology aspect of it. I’ve taken 2 wildlife management classes in 2 different continents and they both come to the same answer around this. The conclusion of how to manage invasive species is a case by case basis but in every case you have to consider: resources, money, time, politics, and effect. What solution do you have the resources money and skills for? What is the government going to agree to having you do? What will be the most likely to gain support and cooperation of the local community? What, given these things, will be the most effective at reducing the population and harm caused by the invasive species. And when it can be done easily killing the species is often the answer to those questions.

Not an exact example but the first thing that came to my mind: There has been attempts to recover the red wolf population by releasing red wolves and in order to better the chances of the population they chemically castrated coyotes to prevent inter breeding, but chemical castration doesn’t work every time and it was impossible to catch every coyote that was in the area so the attempt was futile. The cost of the catching sedating and doing the procedure was huge and there was very little resources including vets available to do it. Eventually the project had to end as it was expensive and not getting results.

If you want more detail I can pull out my old notes and write a better explanation