And it'd require another processor, an arguably unneeded expense.
This is the real reason. The GB and GBC used a z80 processor, while the GBA used an ARM7 processor. The GBA and GBA SP actually contained both processors in there, and the shape of the cartridge flipped a switch to tell the GBA which mode to boot in. When the DS was designed (and the Game Boy Micro for that matter), they had to cut corners to get everything to fit, so the z80 was removed.
Also, keep in mind the DS was supposed to be a "third pillar" on its own aside from the Gameboy line. The addition of GBA support was likely added to give people an incentive to buy the strange new device without making the upcoming Gameboy system (whatever it would be) entirely redundant.
If you wish to play classic GB/GBC games on your DS, you'll need to buy some homebrew equipment. If you get a slot-2 device combined with a slot-1 passthrough, you can use Goomba Color, Lameboy, or DSBoy, while if you choose a standalone slot-1 device, you can only use Lameboy and DSboy. Honestly DSBoy isn't really worth using at the moment, though in the future it may get better. Right now the main contenders are Lameboy and Goomba Color. Both of them have different levels of compatibility and speed for various games, so it might be best to get a slot-2 device simply so you can have the greatest compatibility. This also gives you the advantage of being able to use Goomba Color on regular GBA systems as well. If you get a slot-2 device, make sure you get one with GBA compatibility or you'll be limited to Lameboy and DSboy only.
Dwedit was working on porting Goomba Color to the DS, but he couldn't get it to compile correctly so I think he gave up eventually.
