A little to add but you have nailed it on the head here.
Elephant hunting is big business for reserves, aside from overpopulation, elephants are generally incredibly pricey to shoot. The money that is collected from this activity not only pays for fences, security and workers to manage the reserves/game farms, but also provides an economic incentive for landowners to keep such animals.
They do cause a lot of problems, be it fence breaks, attacking vehicles and people or tree damage which is the most noticeable and common form of damage.
If you ever go to the Kruger you will notice a big difference between areas that have elephants and those that do not.
If it wasn’t for hunting packages you would see far less elephants outside of the national parks, which up until recently the Kruger had yearly mass elephant culls. I’ve been there a lot in the last year and can say with confidence that I’ve seen more elephants than impala most days I’ve driven through.
In parts of Mozambique that I’ve visited they’ve caused havoc to farms and community’s, killing people unprovoked and destroying crops and irrigation equipment.
Recently Botswana has actually criticised Germany for placing a ban on elephant trophy hunting for its citizens, and threatens to send them 10,000 elephants for them to deal with!
https://youtu.be/7uvz62-AenA?si=fDOxhZ4iuPdBOhlE
Elephant hunting is legal in most large number of southern African countries and encouraged.