I think what you mean when you use PR-narrative is more like projected brand, rather than narrative (in the PR sense). Narrative (to my understanding and how I use it professionally) is generally limited to a topic, a situation or an event, as opposed to the whole of the organisation. Practically, an organisation might have multiple narratives on different topics.
Another important distinction between the strategy and brand, I think, is strategy is more prescriptive and brand is more interactive. Yes, strategy is informed by the environment, players, etc but then describes a situation, includes predictions, identifies challenges and defines actions to overcome those challenges. The brand is more interactive because even though you can project image and morals, in the end, the impression of your actions on your target audiences and their reactions, therefore your reputation also plays a huge role in that.
You always want to control your brand (and most of the time the narrative in a situation) to create an advantageous position. But it might not always be the case. And yes, the disconnect between the actions and the brand might create tension that might damage reputation if allowed to grow.
Cool. Sometimes people use “meta-narrative” for that kind of thing if I am understanding your point correctly. Like the overarching message-focused story of an organization. But, sure, use narrative, as long as people understand what you mean, all cool.
Yeah, your observation is on point—most of the time people (audiences) do not actually “track” about an organization. But that doesn’t mean it does not matter—on the contrary. The narrative is generally absorbed subconsciously, by being exposed to multiple stories from the organization.
I’d say the mission statement is something else than the narrative and more of a part of the strategic-domain than communications-domain. But definitely, the mission informs communications.
On your last point, I’d not say “less coherent” but maybe “vaguer”. If you are trying to control a narrative, I wouldn’t say it is not coherent. It is hard to control, yes. But I am not sure if it lacks coherency. But I think, both strategy and narrative are hard to understand when you are looking outside-in but feels more coherent from inside-out.