Sometimes rivers dry up. But generally most rivers (and waterfalls by extension) stay fairly constant because, yes, the precipitation maintains the water flow. If that wasn't true, then the water would just be magically appearing, which I can assure you it is not.

Waterfalls remain active through a continuous supply from their upstream drainage basin, which is fed by precipitation, snowmelt, and groundwater. Most permanent waterfalls are fed by a river or stream that collects water from a large surrounding area; as long as it rains or snows in that catchment area, the river continues to flow over the precipice. In 2026, many of the world's most famous falls, such as Niagara or Victoria Falls, also rely on a "recharge" from massive lakes or underground aquifers that act as natural reservoirs, smoothing out the flow during dry spells. In some cases, such as Bridalveil Fall in Yosemite, the flow may diminish to a trickle in late summer if the snowpack is depleted. However, for major falls, the combination of a vast watershed and the slow release of groundwater ensures that gravity always has a steady supply of water to pull over the edge.