oh yeah, it would sound alot better depending on speakers and amp, but i would just run them through the stereo if they are standard door speakers, especially if you have uprated 6x9s and a sub in the back.
im still sure it would have been the amp connected to the front output leads of the stereo, not the speakers.
have you had another look at those wires that run back under the console

the blue and white wires that are twisted together have got more speaker cable connected which would have probably fed back to the amp. the red and brown wires look the same but cant quite tell from that pic because they dissapear under the carpet.