I work in a similar team and the way it has been 'resolved' is that
We are all responsible for updating the health systems and databases. Funding is provided to the team based on the statistics that are collated there so that is the Trust's priorities.
After much argument, an admin worker has been employed to input to the local authority database for all members of the MDT EXCEPT social workers who are expected to do this themselves.
I don't do too badly on it because I'm quite used to computer systems but some of my colleagues struggle. It's a lot of extra time to put into a job that is already struggling under a lot of time pressure.
As for becoming generic mental health workers, that's definitely happened where I work - all care coordinators, in effect, also care manage.
And a different argument perhaps!