Differential forms inherently measure orientation. The value of a differential form $\omega \in \bigwedge^n(M)$ on an $n$-parallelotope, i.e. $\omega(X_1, \ldots, X_n)$, is interpreted as the oriented volume of the parallelotope spanned by $X_1, \ldots, X_n$. The orientation is a necessary part of the interpretation, since differential forms are alternating:
$$
\omega(X_1, X_2, \ldots, X_n) = - \omega(X_2, X_1, \ldots, X_n).
$$
Hence, when working with differential forms, we should expect things to go wrong if we throw the notion of orientation out the window.
Concretely, say we want to integrate a differential form $\omega$ over the image $\phi(U)$ of a single coordinate chart. Say that $\omega$ is written in coordinates on $U$ as $\omega = a dx_1 \wedge \cdots \wedge dx_n$. The usual way to define the integral is by pulling back into Euclidean space:
\begin{align*}
\int_{\phi(U)}d\omega
&= \int_{\phi(U)} a \, dx_1 \wedge \cdots \wedge dx_n \\
&= \int_U (\phi^* a) \, dx_1 \cdots dx_n \\
&= \int_U (a \circ \phi) \det d\phi \, dx_1 \cdots dx_n.
\end{align*}
This final expression involves the Jacobian determinant of the coordinate transform --- the factor picked up by change-of-variable --- and its sign depends on whether $\phi$ is orientation-preserving. Hence, if we flip orientations, we flip the sign of the integral. Hence we must have an orientation on our manifold in order for integration to be well-defined!
Clearly, if we want to be able to integrate without worrying about orientation, we either need to
(a) change the definition of $\int_{\phi(U)} d \omega$, or
(b) integrate against something besides differential forms.
It seems you are arguing that we should try (a). But as long as you want your definition of integration to make any sense (e.g. be independent of things like choice of charts or partition of unity), by pursuing (a) you'll probably end up arriving at something that is morally more like (b), since we invented differential forms to be orientation-measuring objects in the first place. In fact, you may end up reinventing the exact concept of density that you were trying to avoid!
On that note, it may comfort you to know that any differential form yields an $s$-density $|\omega|^s$ in the natural way, as
$$
|\omega|^s(X_1, \ldots, X_n) = |\omega(X_1, \ldots, X_n)|^s.
$$
So, moving from differential forms to densities is really quite natural --- they're just an orientation-forgetting generalization of differential forms. The machinery involved in defining them is a bit more complicated, but that's the price we pay for dropping orientation.
Looking at it another way, it may be helpful to replace the word "possible" in your question with the word "useful". After all, any construction (that is not logically inconsistent) is possible in mathematics, but most constructions are not useful. Making that substitution:
In other words, is it really not useful to define integration on nonorientable manifolds using forms alone?
No, it's not particularly useful. See above --- orientation is baked into the definition of differential forms and their integration. Attempting to wrangle forms into playing nicely with orientation-less structures won't be pretty. We'll cause a lot more problems than we solve by trying to do that. If we want to forget about orientation, we should integrate against something else.
I've seen some discussion of using "densities" instead of n-forms for integration, but am not really clear on why densities are useful.
Densities are useful precisely because they solve the problem we're talking about here --- they are the closest thing to differential forms that we can integrate without having to worry about orientation.
I hope this clears things up!
