I'm not sure why the transporter is so difficult to accept within the context of established Treknology.
Trek has told us more than once that in the 23rd/24th century, matter and energy are interchangeable. They can be converted back and forth at will. Therefore, when you are transported, nothing is destroyed. You are simply converted from matter to energy, the energy is sent to the destination, and then you are converted from energy back into matter.
Now, that may not make much sense for us today based on our scientific knowledge and level of technology. But it is apparently true within Trek.