Wearing a morph suit would protect her identity. The only thing that really hides a costumed person’s identity is sheer anonymity of population. If you’re not a public personality no one will ever think twice about stranglers out of uniform. Only risk is friends and family seeing likenesses, that’s why you keep your vigilante identity out of the media.
Batman kind of gets away with it because Batman is traditionally reclusive hero despite the high profile Bruce Wayne, but those close to Batman should have a suspicion of his true identity.
Ollie is another level of ridiculousness. The goatee should have been a fake beard to hide his identity, not his actual facial hair. It only made sense that his identity be made public.