Unfortunately yes they do get removed 😭
A graveyard in Yorkshire that has a large amount of my direct family, removed the majority stating they were dangerous !!!
3000 graves on the church yard going back to 1630 and now only a few hundred remain.
I have photos of the graveyard full of beautiful stones but now its just a field inside the church yard wall. its breaks my heart.
Do the council smash them up and use them as aggregate etc 
You seem to be talking about two different things here, churchyards and cemeteries, they come under different jurisdictions and regulations. The following apply specifically to CoE churchyards but other denominations have similar regulations.
Churchyards are subject of rights and obligations affecting numerous people. Some of the laws applies to land similar as to other land, some specifically to church property and some uniquely to churchyards. There is no right of way in consecrated ground unless the dedication or grant is authorised by a faculty (though a path used over a long period of time may be presumed of a faculty that has been lost).
Grave stones belong to the persons commemorated in in turn to their heirs in law, it is their responsibility to care for them and keep them in a safe condition.
A faculty is required to remove gravestones and under current law if a gravestone is removed and the current owners do not want them or cannot be traced the stone must be broken up.
Many churchyards removed the majority of their gravestones in the early 1960s when mechanical mowing became popular, the gravestones were placed around the churchyard wall, leaving only a few in place.
A later "health and safety" cull of stones took place circa 2017 to 2020 after a tall stone fell on a child.
Cemeteries are subject to the control of the local council who may pass their on by-laws covering the cemeteries. Many cemeteries were subject to very heavy handed "health and safety" cull of stones circa 2017 to 2020, some even removed curb stones in case people tripped over them.
Cheers
Guy
