Mz: "So basically, alligators dig holes, and this does a bunch of important things, and also they eat one of the main animals that would otherwise destroy wetlands."
Well, that's not what you originally said. Furthermore you're misinterpreting what you quote above too.
FWIW in the U.S. the word "coypu" is rarely used, we use the word "nutria" instead.
First of all, to say
Mz: "...like an alligator making swamps in the American South..."
is like saying that cows make cow pastures, or prairie dogs make grasslands, or oysters make the seabeds that they lie in - it is essentially incorrect.
I lived decades in the American South and have seen my share of alligators. I even raised one awhile and she did not terraform the backyard in any manner whatsoever. Mostly she crapped, irritated the dog, hissed and stood her ground until fed. To read someone speaking of alligators "making swamps in the American South" still has me laughing.
Mz: "Removing alligators from certain swamps would, in fact, significantly impact the swamp and make it much more likely to revert to solid land over time."
No. The swamp would be there with or w/o alligators. For example, Disney is removing alligators from their Disneyland parks. The swampy areas won't dry up as a result.
Swamps are consequences of geography and climate, not of alligators. There are plenty of swamps w/o alligators and alligators w/o swamps (e.g., lakes, rivers, etc.).
Humans are probably the main predators of nutria. Humans introduced nutria to the U.S. for their fur, in some instances, and to control various wild grasses, in others. Were the alligators were all gone, I have no doubt that other predators would easily pick up the slack.
Even if the nutria were allowed to reproduce unencumbered, they would not "destroy wetlands". Firstly the population would, as in the usual predator-prey models, boom then bust. Secondly, as I said earlier, the wetlands are a product not of biology, but of geography and climate. Of course biology diversifies the environment, but it doesn't create it. You could hit Louisiana with a giant neutron bomb, killing every living thing there and the wetlands would remain for eons.