Let the math to the math guys who never ship a product.
Re-writing recursive algorithms to be non-recursive typically requires less obvious code, where you make your own ad-hoc stack to keep track of where you are; essentially implementing the call stack manually.
In contexts where DoS or stack smashing is a concern because the input is attacker-controlled, it's often way easier to add a depth parameter and error at a certain depth than to rewrite the naturally recursive algorithm into iterative style. But tree structures are so naturally recursive that it's easy to end up with accidental unbounded recursion in complex real-world situations.
You can manage memory and compute budget so that your cute algo does not go rogue. On top of that very often you don’t have to explore the entire tree, and there is custom logic at each step to decide whether you want to proceed or not with exploration.
Example: depth-first tree walking algorithms. Implicit stack makes it trivial to express as recursion.
It is not smart, or special, or something.
The secret is to check at every step that there is no "he's his own grandpa" loops in the user's tree, where (s)he inadvertently makes one of a person's descendants, also his ancestor. This happens sometimes because re-using the same names can cause confusion.
Recursion is like magic :o)
I don't think we should ban recursions altogether, but remember that there exist associated risks, and consider them.