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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?

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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.

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