I know that repeaters on VHF and UHF are primarily automatic, but there will be times when they will fail, or something is not working. The repeater may not respond to the remote control. In these situations, someone will have to climb the top of a building or go up to the top of a mountain and tinker with it. Who can do this?
The owner of the repeater is typically in control of who has physical access. Often the owner will permit the trustee of the station license (in the US) to have access in order to carry out their lawful responsibilities although the owner and trustee may be one in the same.
When a club maintains a repeater, there is often a technical committee that is charged with keeping the repeater in good working order.
Also consider that in the US, from a legal perspective, the control point of the repeater may be remotely located from the repeater even if the repeater is normally automatically controlled.