Well, I ended up finding how to reset the SelectedItem of the picker:
first of all I make sure of the position of the picker:
PickerMoment.SelectedIndex = 0;
Now that I know the value displayed, I manually initialize the selectedItem (which is rather unusual). As I know the initial value this is not a problem.
PickerMoment.SelectedItem = "Indifferent";
what
is surprising is that Android has no initialization concerns, it can
directly read the selected item, for iOS the selected item is only
initialized if its value changes during use ( in
other words, by default the selected item does not initialize, but if
you swipe on the next or previous iOS initializes the item)