Lattice Crypto worst case to average case
You are right.  A worst-case to average-case reduction from problem P to a distribution D over instances of problem Q would mean roughly that $$\Pr_{q\leftarrow D(Q)}[q\text{ is hard} ~|~ \exists~\text{any instance of P that is hard}]&gt;1-\text{negligible}.$$ So even if the support of distribution D is the entire set of instances of problem Q, this does not of course imply that every instance of problem Q is hard.

As an aside, it should be pointed out that the way in which parameters are set for lattice problems does not result in a particularly meaningful worst-case to average-case reduction because the reduction is not tight -- the dimension of the lattices in problem P is smaller than in problem Q.  

Still, the big utility of having had such a reduction was that it was this reduction that led people to the problems of (Ring/Module)-SIS and (Ring/Module)-LWE and distributions over these problem instances (uniform) that are currently used in practical lattice cryptography.  Analyzing the hardness of random instances of these problems became an independent matter.