Resolves to such problems depend on many factors, such as management and other staff skills and attitude, budgets, available staff time, etc. As such, this post is just ideas based on a general workplace but also relevant to what you reported.
A core issue that must be resolved is that person X is currently seen as a problem, and your reporting this to others and management may make them feel you are a problem too. No offence intended, just being factual. This makes bad vibes in connections between multiple colleagues, and thus the group/dept as a whole.
I know this issue is forced by the specifics of the scenario, but this is something you need to resolve, otherwise your dept wont work well and wont get the resolve it needs.
New recruit option.
A "dedicated PR/marketing person" is only required when the current staff/team cannot manage the task(s). Either because:
- There's a level of experience required which is not currently present
- Due to time constraints
If this is the case then this would be the ideal course of action you need to take. Once someone dedicated and able to carry out these tasks is in place, they should resolve the issue of X feeling they have a task that's not their responsibility, and the problem that this all affects the dept as a whole and how it is perceived.
Another option - including if, for whatever reason, no additional staff are to be recruited.
One core issue I get from your post is that the roles and responsibilities are a little loose. I think your first priority is for whoever is in charge to make it clear and official who is responsible for what tasks.
However, that doesn't need to be a bad scenario of people being told to "just get on with it", or single responsibility.
When people look at us closely, they say our brand is unclear, mission
is vague, webpages are confusing, & business model is hard to discern.
You use the words "our" and "us" - you are a group and while most times individual people have their own specific tasks, some things need or benefit from a group task force, or a group to discuss things even if one person has sole responsibility.
One approach (if plausible) is to form an official group. Speak with whoever is the authority that can form a group, and discuss with them the issues you mentioned and you think it could be better resolved with a collection of minds.
Then once a week (whatever frequency is needed), get together and discuss the problems the dept faces. This approach may also resolve individual's issues because a group alleviates issues caused by one person having too much work, and the pressure they may face from single responsibility.
X might feel less pressured when other people share the responsibility, and you might find X actually has some good suggestions.
The first things the group should discuss are as quoted above. You need to discuss and figure out what your core mission statement is and your core values as a dept. Once you have that you can determine what your business model should look like, and then from there determine what your brand is to resolve it being unclear.
Once you have a good structure as above, you know what your core values and mission statement is, you can discuss and determine other things. Such as what you will market and how, and how PR will look, and who is responsible for these things.
This may not be feasible, as generally resolving these issues should just naturally be done by management, and if they haven't already then perhaps the issue lies there. They're not taking control and implementing necessary protocols and changes as required. Maybe the group thing will help them too, figure out where everyone is at, what certain core things the dept has, and so see clearly what the next steps will look like.
If a group is not feasible, then the above can still be used as a basis by whoever is responsible. Instead of discussing the issues they just need to think about them and propose solutions to whoever can authorise and make changes.