For a virus to be the most "successful" it needs to keep the host alive! It is not in a virus's best interest to kill its host, or it loses its ability to spread. When a zoonotic virus jumps species, it may inadvertently kill a number of the new hosts before it has time to adapt to the new species. Those hosts that survive spread more viruses than those that die. All other things being equal, a strain that kills fewer hosts will be more successful than a strain that kills more hosts.
Think about how successful the rhinovirus that causes the common cold is. It rarely kills the host, only slightly incapacitates the host (people often keep going to work and to school when they have a cold), and has infected an enormously high percentage of the population at one time or another.