The heuristic fails for precisely the reason you state; there is a parametric family of solutions which makes it fail.

People often use heuristic arguments to predict the number of integral/rational solutions of bounded height to diophantine equations. These heuristics are often true, provided one allows a small margin of the error given by ignoring possible parametric families of solutions which make things go wrong (often called "special subvarieties"). This is exactly what is happening here.

This philosophy is present in many conjectures and research problems in the modern study of diophantine equations, such as Manin's conjecture, the Bombieri-Lang conjecture, the André–Oort conjecture,.....