The integration sign in $\delta I$ is there because the integration sign was there in the original $I$ to start with. The term "variation" means the addition of $\delta$ in front of the object. It means we study an infinitesimal differential of the object; the rules that obey the variation are identical to those for other derivatives including, for example, the Leibniz rule for the variation of a product.
The only possible way how the integral sign could disappear would be if we were taking the derivative of the function $I$ with respect to $t_2$ or $t_1$ (the upper or lower limit; the lower limit would pick a natural minus sign). But the variation isn't a derivative with respect to a particular variable such as $t_2$, the upper limit. It is the object that knows about the derivatives of $I$ with respect to everything that can vary. The main thing we want to vary are the values of $\delta q(t)$ for any value of $t$, not just a single upper limit $t_2$.