Where the terrain, vegetation type, maintenance budget etc permit then this is an understandable approach, goal maybe. To paraphrase a quote "looking for golf balls is no fun" and wastes time.
However, there are alternatives one of which is designated 'ecology areas' on courses. Some are backed-up by regulations, some are voluntary.
I've played a few UK courses with these designated areas and the premise is certainly not 'no mowing', rather it is let rough grass away from the usual lines of play grow through the spring and most of the summer and then crop it, harvest it even, in say August. This allows wildlife to nest, breed etc in the longer grasses but by cutting it once per year scrub and trees are unlikely to sprout high and grow tall.
Another approach, where there are no other animals around that are likely to eat them, is grazing sheep, cattle, horses and goats and goats in particular will eat most vegetation.
atb