Salt solubility and solvent dissociation A textbook, describing a qualitative test for the $\ce{Al^3+}$ cation in which you use $\ce{NH4Cl}$ and $\ce{NH4OH}$ to precipitate $\ce{Al(OH)3}$, mentioned that the solubility of this precipitate decreases in the presence of the ammonium salt.
I sort of reasoned it out to the common-ion effect decreasing the $\ce{OH-}$ concentration before I wondered why that would have an effect at all. As far as I know, the solubility of a salt only varies with the solvent used, the pressure, and the temperature (and of course, on the common-ion effect if it's present).
I made up an explanation, that a more dissociated solvent wouldn't have to physically contort to solvate the salt ions; that when the bonds are broken, the ions of the solvent can more freely arrange themselves in optimal positions to stabilize the salt constituents.
Is that line of reasoning correct? If not, how can the relation between solvent dissociation and salt solubility be rationalized?
Common ion effect is the key.
A solution of $\ce{NH4OH}$ and $\ce{NH4Cl}$ is basically(no pun intended) a basic buffer solution, which has an almost  constant $\mathrm{pH}$ and thus a fixed concentration of hydroxide ions.
The solubility product of $\ce{Al(OH)3}$ as you say is also constant. So as to maintain the constant nature of both $[\ce{OH-}]$ and $K_{\textrm{sp}}$, there will be a fixed equilibrium concentration of $\ce{Al^3+}$ for a given $\mathrm{pH}$ above which it will start precipitating as $\ce{Al(OH)3}$. By increasing the hydroxide concentration we are forcing the $\ce{Al^3+}$ cation to precipitate earlier than it would have done in a normal, neutral aqueous solution.
This process of forcefully changing the $\mathrm{pH}$ of a solution to alter the solubility of different compounds is very common. In this case it was used to reduce the solubility and precipitate a substance, in other case it might be used to increase the solubility.
For example, during the same qualitative analysis schedule you mentioned, $\ce{HCl}$ is added alongside $\ce{H2S}$ during the detection of group 2 cations(like $\ce{Cu^2+, Hg^2+}$) to prevent precipitation of group 4 cations(like $\ce{Zn^2+, Mn^2+}$) in the form of sulphides. As the acidic conditions increase the solubility of sulphides, added with the fact that group 4 sulphides are not that insoluble prevents them from precipitating with the group 2 cation sulphides.