The context
I am studying constrained systems in Classical Mechanics following the book Anaytical Mechanics. The autor considers a system of $p$ non-holonomic constraints that have the form \begin{equation} \sum_{k=1}^na_{\ell k}(q,t)\dot{q}_k+a_{\ell t}(q,t)=0,\,\,\,\ell=1,\dots,p\tag{1} \end{equation} He then proceeds to write this equation in the equivalent form \begin{equation} \sum_{k=1}^na_{\ell k}(q,t)dq_k+a_{\ell t}(q,t)dt=0\tag{2} \end{equation} Now, he takes into account the fact that the virtual displacements $\delta q_k$ have to be compatible with the constraints with fixed time, and so he sets $dt=0$ and gets the equation \begin{equation} \sum_{k=1}^na_{\ell k}(q,t)\delta q_k=0\tag{3} \end{equation} Finally, he multiplies the last equation by the Lagrange multipliers $\lambda_\ell$, sums over $\ell$ to $p$, integrates, and obtain the variational principle \begin{equation} \sum_{k=1}^n\int_{t_1}^{t_2}\left[\frac{\partial L}{\partial q_k}-\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_k}\right)+\sum_{\ell=1}^p\lambda_\ell\,a_{\ell k}\right]\delta q_k\,dt=0\tag{4} \end{equation} which yields the Lagrange's equations.
My background
I understand the idea of thinking about the virtual displacements being tangent vectors in the configuration space and this sounds more intuitive to me than "a displacement that occurs with fixed time". However, I do not understand how to derive equation (3) using equation (2) in a more rigorous way (although I get the author's argument) because I am not really familiar with differential forms, if that's even what equation (2) is about.
The Question
I would like to know if there's a way to get equations (3) and (4) directly from equation (1) treating the virtual displacements only as tangent vectors, without using equation (2) and setting $dt=0$, if that's possible.
I have seen the answer to this question, and I think the answer to my question would be something similiar to consider a new parameter $s$ rather than $t$, but I do not really know how to formalize this.