To fully answer this question requires all sorts of background information. I think I will start with a couple of things that I think you may have misunderstood.
The SD card (or a small USB stick) is called a "Flash" drive because that is the name of the "Flash" type memory it has in it. When you copy a file to an SD card, the circuits inside it move charges around the chip in a way that will persist when you unplug it. (Ordinary "Dynamic RAM" memory chips in a computer lose everything quickly as soon as the power goes off.) Flash memory is very slow to write to, and slow (but not as slow) to read from (compared to Dynamic RAM). Flash RAM also "wears out," so after thousands of read/write cycles parts of it quit working.
Fortunately for you, most of that is invisible in practice, so you can pretend an SD card or Flash USB stick is very similar to a tiny hard drive or floppy disk. But both are quite different from "burning" a CD or DVD.
SO try to forget about "flashing" the information to the SD card. What you are doing is not exactly comparable to "flashing" a computer motherboard or some other piece of electronics, because you don't have to go to special effort to change the data in the SD card's memory. Deep down on the chips it might be exactly similar, but the process you use write to an SD card or USB stick is quite different.
The Raspberry Pi has been designed to look on the SD card for a boot partition using a FAT filesystem. Depending upon how you are using the Pi, that might be the only partition on the SD card, but a sophisticated operating system such as Raspbian cannot possibly run on a FAT filesystem, so you need at least one additional partition (or a separate drive) that can handle linux files, such as ext4 format.
When you are running linux or Windows or a Mac and "copy" a file from one "mounted" filesystem to another, the operating system takes care of creating, deleting and pointing to the files as they are put on the hard drive, flash drive, or whatever. The files get stored inside each partition using the rules specific to the filesystem, whether it's FAT, ext4, Mac HFS+, or whatever.
When you start with an image file (often identified as .iso for linux or .dmg on a Mac), the file contains EVERYTHING, including all the pieces that identify the partitions and the chunks of data inside the partitions.
When you use the dd command in linux to copy a file, it takes the bits out of (or into) an .iso EXACTLY as they are stored on a disk or an SD card, completely ignoring the partitions, filesystems, files and directories that you and the operating system need to make sense of the data on it. It's a completely different kind of copy operation. If you just drag that .iso to your SD card, not only will you needlessly fill it, the Pi doesn't know how to look "inside" the .iso file to get what it needs.
A dd copy takes a long time because SD cards are slow and you are copying large chunks of data at a time. When everything is accounted for, however, it might be faster than copying the individual files.
And to answer the question, yes, you could mount the image and copy the files into the various partitions, but there will almost always be more than one partition and you will need to put the files in the right places. It's easiest in many cases to just use dd to copy from the image files to the SD card.