What Eckels's action shows is that one small change can have huge consequences. The change in the election outcome shows the reader that something as seemingly mundane as a single butterfly death can change the very organizational structure of a country.
Regarding your second question of whether or not I believe something like that could happen, yes, I believe that it could happen. What Bradbury builds a fun story around is the concept of "The Butterfly Effect." It basically says that one small change can have huge impacts on a lot of other things.
Interestingly, Bradbury is not the man credited with coining this idea. That recognition goes to Edward Lorenz. He was a meteorologist in the mid 1900's. Like all meteorologists, he was working with various weather models and changing data sets to see their effects. In one such simulation, he changed a minuscule amount of data and discovered that the weather outcome was completely different than the original. Lorenz then made the interesting connection that a single flap of a butterfly wing could potentially change the outcome of an entire weather system. He called it "The Butterfly Effect," and the name stuck.
His data driven model supports the idea that small changes can equate to massive changes later. That's Bradbury's story, so yes, I believe Eckels's actions could have had the narrative results.