The problem is you pass RHF's onChange handler to the wrong prop. <ReactFlagsSelect /> doesn't have a onChange prop, instead you should pass it to the onSelect prop.
<Controller
name="country"
control={control}
render={({ field: { onChange, value } }) => (
<ReactFlagsSelect
selected={selected}
onSelect={onChange}
value={value}
/>
)}
/>
Side-note: RHF's will update value changes to your registered fields in it's internal form state, so there is no need to use extra state management for these values via useState or something similar. If you really need to call your handleChange callback, you can do both in the onSelect callback.
<Controller
name="country"
control={control}
render={({ field: { onChange, value } }) => (
<ReactFlagsSelect
selected={selected}
onSelect={code => {
onChange(code);
handleChange(code);
}}
value={value}
/>
)}
/>