Indeed, the transporter is one of the few Star Trek concepts that showcases the limitations of our caveman brains - and quite accidentally at that, as this thought-provoking technology was a last-minute addition to the show's concept. Warp drives and death rays may invoke their own little tempests in scientific teacups, but it takes some doing to see the philosophical implications of those. With the transporter, the ontological oddities slap us in the face.
I think your chances to die in a car accident today are greater than to die in a transporter accident in Trek's future.
...I wonder what sort of statistics LaForge was referring to when assuring Barclay that transporting was the safest way to travel.
Accidents per kilometer? That's a classic way to evaluate traffic safety, since covering distances is a typical purpose for traffic. Thus, air traffic today gets high marks. But transporters generally probably see very short distance use, from spot A on Earth to relatively nearby spot B also on Earth, even if Starfleet sometimes uses them at ranges of tens of thousands of kilometers. Transporters would be ideal for commuting, removing the vehicle from the equation and thus decreasing congestion and serviceability problems.
Accidents per person? There are more people in the UFP than on Earth, but we have no good idea whether a greater or lesser percentage of UFP citizens use the transporter than Earthlings today use, say, a car or a plane. So, accidents per person actually using the device? Transporters have fewer modes for killing non-users, the way cars kill bystanders, so this sort of comparison is sort of unfair.