Never seen anyone injured himself doing a handstand. But running has a high injury rate and still a lot of people find benefit practicing it
But if you do it safely, risk of injury is very low. Less than spraining an ankle running on wet grass (which football and soccer players do all the time).
the load on the shoulder joints in this position is very unfavourable.
millions of people run every day, of course there are injuries.
It’s the finger, wrist and elbow joints and ligaments (your fingers need to act as toes, your arms are now legs) and not learning how to bail safely (if you flip over and don’t have the reflex to cart wheel out).
There's a reason why handstand pushups (often done against a wall for muscle-building/strength purposes) are considered an advanced shoulder exercise. It's sort of the calisthenics equivalent of the overhead press.
Do free floating handstands for time and see what hurts more.