Why do we only have one major meteor shower from 3200 Phaethon?
Note the short straight lines that hang down (or stick up) from the orbital track of 3200 Phaethon you reference.
These indicate the distance above or below the plane of earth's orbit (the ecliptic) 3200 Phaethon is at that point.
So 3200 Phaethon passes well above Mars' orbital track and passes through the Earth's orbital track (just about)
Then it passes under Venus, under Mercury, makes a U-turn around the Sun, and passes back over Mercury, Venus Earth, and Mars.
So your supposition in the last paragraph is correct.