Well, first you are assuming that texture compression has come nowhere since 1998. .
Actually, it didn't improve much. The major improvement was hardware-based compressed formats like S3TC (which the 3DS uses), which can approach 24-bit quality with 8-bits or even 4-bits per pixel. On top of that, games can use lossless compression on the files themselves (zlib, lzma, etc), but this is only used if really necessary since it increases load times and uses more RAM (decompression buffer).
Nobody uses stuff like JPEG for textures, except for browser-games, since it produces terrible results and uses far more RAM than S3TC, which can be used directly by the GPU without decompressing first.
Second, even if all textures were 16 times larger, the game still would not be 16 times larger as a whole because textures make up only a portion of the game's size.
Sound effects may be higher bitrates, but I would not say it is obvious because it sounds almost identical to the original, aside from the improvement in the few vocal samples the game contains. The same goes for the music, which has no decipherable improvement.
The FMV explanation makes more sense in explaining the huge increase.
Outside of video and streamed audio, textures almost always take the most space than anything else in any game.
Question for smart people: Super Mario Galaxy rips to about 3 GB, a bit over. Super Mario Galaxy 2 rips to half that. Did they massively improve their compression? What's up with that? Not to mention that Galaxy 2, to me, looks more consistent overall (less randomly blurry textures here or there).
At least the intro and the ending in SMG1 are actually 60fps FMV. Those might take quite a bit of space.
There is that. The larger the ROM, the slower the seek times. That's just the physical nature of it. Also the DS/3DS carts only have about half the physical contacts of a GB/C/A cart (and therefore only half the bus width).
Solid state media doesn't have "seek time". There's nothing moving anywhere to "seek" anything. And the number of contacts has little to do with bus width, otherwise USB 2.0 would be as slow as USB 1.0.
The difference between old-style carts and DS/Vita/3DS cards is that the former are mapped as memory (the cart is a chip plugged directly onto the console hardware, and the CPU can see all of the ROM at once) while with the latter the console can only see a block at a time and must tell the card to switch to a different block in order to get more data (this process can be automated via DMA to copy data to RAM without CPU interference), which is similar to how a PC reads data from any kind of storage (HDD, flash, optical, etc).