As action is defined as
$$S = \int_{t_1}^{t_2}{\mathcal{L}(q,\dot{q},t)}dt $$
For any time interval $(t_1, t_2)$.
As $t_1$ and $t_2$ are arbitrary $t_2$ can be taken arbitrarily close to $t_1$ and we could drop the integral sign.
Why isn't that the case?
Well, it is possible to work directly with the Lagrangian in Lagrange's equations, cf. e.g this Phys.SE post.
However, if one wants to have a variational principle that leads to Euler-Lagrange (EL) equations, it is necessary to introduce the action functional $S=\int_{t_i}^{t_f}\! dt~L$. This leads to the principle of stationary action/Hamilton's principle. Recall that if the Lagrangian $L$ depends on generalized velocities $\dot{q}^j$, it is necessary to integrate by parts to derive the EL equations. If one drops the time-integration in $S$, as OP suggests, this derivation becomes impossible.
