The key fact is here that a [virtual displacement](http://en.wikipedia.org/wiki/Virtual_displacement) $\delta$ only affects the generalized coordinates $q \in Q$, 

$$ \delta q ~=~ q_1 - q_0. $$

It does by definition not affect the time variable $t\in[t_i,t_f]$,

$$\delta t~\equiv~ 0.$$

In other words, a virtual displacement always refers to the _same_ time $t$.

Therefore if one would like to view the virtual displacement $\delta q$ as a [homotopy](http://en.wikipedia.org/wiki/Homotopy) $H: Q \times [0,1] \to Q$ with

$$H(q_0,s=0)~=q_0\qquad\text{and}\qquad H(q_0,s=1)~=q_1,$$ 

where $s\in[0,1]$ is the homotopy parameter, then one can _not_ identify the homotopy parameter $s$ with time $t$.