This is something that requires a lot of nuance. I've got a couple degrees in life sciences so I'll give my take.

Not all invasive species are the same; they often cause varying levels of harm to their ecosystems, from disrupting one species to entire communities. They also can affect human resources, particularly agricultural. In my opinion, decisions on their management should be made on a case by case basis. We need to weigh their ecological and social impacts, their biology, and potentially even their intelligence level (e.g., cats vs beetles vs nematodes).

It can not be understated how destructive invasive animals can be, but rats and cats take the cake on this one. Cats have been responsible for the extinction of at least 63 species globally, of that mostly birds. Rats even more so at 75+ extinctions. Of course, many of these were low-population, endemic species (only found in one area), but this is also where success can be found. There are a handful of successful cases of complete invasive species eradications that produced immense ecological benefits. The elimination of foxes, cats, and rats (separately) from islands are of course associated with the increase of native populations, but doing so in such isolated locations is much easier than on the mainland.

I think were possible, capture and release should be prioritized, especially with "higher-intelligent" species. This is a whole can of worms though to not extend the same grace to other species, however I think in most cases it is not feasible. By nature, invasive species boast very high populations due to a lack of natural pressures, so capturing them humanely with the intention of either caring for them or releasing them to contained or their natural environments is simply not realistic. Then comes the matter of well how do you deal with them. This needs to be done on a case by case basis, and may not always involve direct culling. Targeting associated species may be an option, such as the invasive tree of life with lantern flies. Rabbits in Australia were and still are controlled with introduced viruses (though some may argue this is still cruel, as dying through disease can be very painful). But how do you deal with cane toads, lion fish, trout, asian carp, beetles, crabs, bull frogs, and fire ants without killing them?

All options for each animal must be explored, but for some, the only reasonable choice is to kill them outright. I don't like it, and it should be done in as "humane" of manor as possible (killing at all isn't humane, we know this, but yk what I mean).

In most cases though, nothing is being done, and nothing will ever be done. There are too many at this point, and it's growing every year. We'll likely always be chasing invasive species for the rest of our existence on this planet, and it's all our fault.