In your metaphor, the question I would ask is whether the wanderings up and down the bank "looking" for a natural crossing are not inherently limited by the probability of the neutral mutations. I hope you would agree that there is some such limit.I'm not sure what limit you mean. An iso-fitness region is, of course, limited. But the limit for possible mutations (and fitness)is potentially so large it's pointless to talk about that. It keeps getting larger (and drives evolution, does it not?).
Perhaps you are unaware of the great potential for functional mutations generated by random change. Think of somatic mutations of the genes coding for immune recognition proteins. Think of the rapid evolution of genes coding for protein coats of viruses.
Limited by time and probability. I would think that the "range" of the river crosser making one river crossing within unit time is a probabilistic cloud centered about the original position. The cloud is "denser" (higher probabilities) in the center and falls off thereafter. With unlimited time, the cloud grows bigger and bigger, but we are not dealing with unlimited time.