Why is the impulse response function of this system 0?

Suppose I have an system $$y(t) = t^{2}x(t)$$.

The impulse response of this system would be: $$h(t) = t^{2} \delta(t)$$.

Since $$\delta(t) = 0$$ for $$t \neq 0 , h(t) = 0$$ for $$t \neq 0$$. And at $$t=0, h(t) = 0$$.

But since $$y(t) = \int_{-\infty}^{\infty}x(r)h(t-r)dr$$, wouldn't $$y(t) = 0$$?

What am I missing here?

This system $$y(t) = t^2 x(t)$$
is not LTI and therefore does not have an impulse response of the form $$h(t) = \mathcal{T}\{\delta(t)\}$$.
So your statement $$h(t) = t^2 \delta(t)$$ is not correct... Hope this solves your confusion.