Why $F(\mathbf q,\dot{\mathbf q},t)$ and not $F(\mathbf q,t)$?

However, it seems to me that since $\dot{\mathbf q}=\frac{d\mathbf q}{dt}$, it's only necessary to specify $F(\mathbf q,t)$ for a complete description.

While it's true that the function $q$ determines its derivative $\frac{dq}{dt}$, it's not true that the value of $q$ at a particular value $t_0$ of $t$ determines the value of the derivative $\frac{dq}{dt}$ at that same value. The Lagrangian has a value at a particular time $t_0$ which is a function of the three numbers $q(t_0), q'(t_0)$, and $t_0$. In particular, it depends on more information than $q(t_0)$, but on less information than all of the higher derivatives of $q$ at $t_0$.