What you are referring to are orphan blocks, which do happen from time to time when miners find a block very close to the same time as another miner. However, both of those blocks usually propagate to the whole network before the next block is found. The nodes just hold onto both blocks until the next block is found and propagates out to everyone. The chances of two more blocks being found at about the same time and built upon different previous blocks is very small, but you still wouldn't have half of the network not knowing about one of the blocks. Both blocks would propagate out to all nodes, and they would keep them both until it became clear which chain would become the longest.

But the miners aren't relying on ALL nodes to do this for them. They are relying on THEIR node, or really their pool's node in most cases, to provide that service. Which is part of the reason pools charge a fee for the use of their node...