Thinking about this, I just realized a simple thing: balance in animal population via starvation might not be as grim as we expect (i.e. it happens completely because the weakest/least fit individuals cannot compete for food and literally starve).
The sexual reproductive system is apparently one of the things that stops working earliest when you're affected by disease or malnutrition. This means that malnourished deers will just start having fewer/no babies while they're being famished.
Still grim, because lots of animals would starve, and still very uncomfortable for the deers which, even if they might survive to "old age", will fail to breed due to starvation... but still better than the alternative in which every baby deer gets born in the world as usual, only to then starve to death while they're still very young.
(Also: many (most?) living things have things that eat them, which is another population maintenance mechanism, and no less grim.)
We can certainly avoid the worst of it - but entirely is more dangerous than none.
There are many natural things that are good. Clean air, unique little ecosystems like GP describes, endless variety -- and we should strive to respect and preserve those, but not because they are natural. Poor animals teeming with parasites, population "balance" maintained through periodic overpopulation and starvation (How do people think it happens? Forest fairies tell the deer how many babies to have?), predators feasting on the organs of their prey while the prey is still alive. All these things are natural but not good and we should not seek to replicate them.