Downwash distribution on basis of Lift distribution
You are confusing downwash ($w$) with circulation ($\Gamma$). As per Kutta-Joukowski theorem, lift per unit span (in the case of a single lifting line) depends on the distribution of circulation along the wing span:

$$L'(y)=\rho_\infty V_\infty\Gamma(y)$$

If you have an elliptical lift/circulation distribution, the circulation decreases towards zero at the wingtip, therefore lift at the tip is zero. Induced downwash, on the other hand, is constant across the span.



Source: Anderson, Fundamentals of Aerodynamics

You may believe that downwash is the largest at the tip due to the presence of wingtip vortices. However, the vortices are better understood as downwash rollover at the wingtips. The vortices themselves do not contribute to lift generation (but unavoidable for finite span).