Why are weak acids not considered good leaving groups?
Leaving group ability is tied to the stability of the leaving group. Acidity is directly related to the stability of the conjugate base. If you're a strong acid, the conjugate base is relatively stable in solution. Less so, if you're a weak acid. So the stability of the conjugate base is directly proportional to both acidity and leaving group ability.

Edit:

$$\ce{HA-&gt;H+ + A-}$$

$\ce{A-}$ is the species we're evaluating as the leaving group. Basically, it's a good leaving group if it's stabilized. If that's the case, the acid dissociation equilibrium also shifts toward products, making $\ce{HA}$ a stronger acid. You should be careful to differentiate between $\ce{HA}$ and $\ce{A-}$ when you're thinking about these concepts.