There are anomalies in what has been going on all these years, because the payment of a 'ground rent' (your term) typically relates to an admission / acceptance that the land is owned by the body to whom the rent is paid to. If the situation is as you say, the council would only expect to be in receipt of money for council tax (if the building was rated for CT).
Paul's answer does tell you how to find out cheaply who the registered proprietor of the land is - if the land is registered with the LR.
If there is no separate lease arrangement for the building (and the LR records do also show up long-term lease contracts) then I'm afraid the building that sits on the land is going to owned by whoever owns the land.
There is no reason for the council to assume ownership of this land - as representives of the community or otherwise - land ownership doesn't 'default' to the council ownership and council-owned land will show up as such at the LR in the same way as anyone else's land would show up. If it is registered land.
The community is likely to have to spend money on a solicitor to unpick this little lot. If it is a community resource, does it really matter? - I guess I am saying, is it worth rocking the boat for, if the community is getting what it wants.