Your component’s async setup() looks fine other than the missing await res.json(), which still wouldn’t cause the problem you’re seeing. I suspect your usage of <Suspense> is incorrect.
To use async setup() in a component, the parent component must use that component in a <Suspense> tag:
<!-- Parent.vue -->
<template>
<Suspense>
<MyAsyncComponent />
</Suspense>
</template>
You could also use the default and fallback slots of <Suspense> to show a loading indicator while waiting for the child component’s setup to resolve:
<!-- Parent.vue -->
<template>
<Suspense>
<template #default>
<MyAsyncComponent />
</template>
<template #fallback>
<span>Loading...</span>
</template>
</Suspense>
</template>
demo