There is a huge difference between normal uae of a garden.
The neighbours garden is much larger and they are planning on using trees right on the boundary, erect a platform which would mean the OP's garden can be overlooked.
I think it is perfectly reasonable for the OP not to want this.
I'm in a similar situation as my garden is 150 ft long and is bordered at the very end onto a garden that is probably max 30ft from the back door.
We also have trees and bushes on the boundary.
We could have very easily put our 14ft trampoline at their end of the garden, but we didn't out of basic courtesy.
We knew 4 children and their friends bouncing noisily on it could be very annoying and spoil their enjoyment.
Lots of people would act similarly.
I value my privacy, and the privacy my trees provide at the end of the garden.
I would definitely understand my neighbours being very pissed off if we used one of the trees as a play area, that would mean our children would be elevated to such a height that they could see into their garden and spoil their privacy.
Not being overlooked at the back of your garden is very reasonably something most people would like to preserve.