Conditions for Complete hydrolysis of salt
The question is, in fact, a good one, but the concept of weaker/stronger is confused - and confusing!

A way to describe the situation, using the terms in the question is this: strong acids and strong bases undergo complete hydrolysis in water. The energy gained from hydrogen bonding with a water molecule is great enough to effect complete ionization in water. Examples would be HCl and NaOH.

Water lies in the middle here, being able to be protonated by strong acids and deprotonated by strong bases. Hydrolysis is a measure of how much the water is protonated to H3O+ or deprotonated to OH-.

There are also materials which ionize in water, but not so extensively, e.g., acetic acid and amines such as ammonia. They dissolve, but do not react so much and thus hydrolyze to a limited extent. Confusing, perhaps, that we call these materials weak acids and bases but they have strong internal bonds which prevent their total ionization. These strong internal bonds mean that if you did form the the conjugate acid or the conjugate base (in a salt), they would react strongly with a water molecule to form a considerable amount of the corresponding acid or base. Thus the salts would ionize and hydrolyze where the original acid or base would not be said to hydrolyze (much).

The "strength" of the cation may perhaps be better described as a polarity, which, if it is a "strong", favors hydrolysis, i.e., complete ionization, meaning it is strongly attracted to the oxygens of water and less so to its own anion. By this terminology, Na+ is a strong acid because it is strongly attracted to water (oxygens) at the same time OH- would be a strong base.

A "weak" cation, like NH4+, would be one which will not protonate water to a great extent. A solution of 1 M NH4Cl will give complete ionization to NH4+ and Cl-, but we do not call that hydrolysis. The ammonium ion is stable (strong internal bonds) and only weakly protonates the H2O molecules around it, so we call it a weak base, with limited hydrolysis.