Yes, your summary is correct. To illustrate how this works in practice, my district council visits all burial areas under its responsibility annually to carry out a safety inspection. They grab hold of each memorial and see if anything moves. If it does, they attach a notice to it informing the owner that it has been declared unsafe and that they have six months to rectify it, or it will be taken down. Where the monument is deemed unsafe, usually the taller structures, a fence is erected around it to prevent potential injury. The district council makes no attempt to contact the owners directly. If no remedial action is taken, the monument is dismantled as far as is necessary to remove the danger and the components are laid out on the grave. Across the district, they do this to dozens of memorials each year and, in total, several hundred have received this treatment over recent years.
Incidentally, I notice that nineteen monuments in your churchyard are either Grade II or II* listed in their own right, so any work carried out on these monuments would require listed structures consent and would need to be carried out by an approved stonemason under the supervision of your local conservation officer. I hope the charity is aware of this.