The process of solving an exact 1st order ODE is the inverse of applying the multivariable chain rule to a function, analogous to inverting the product rule to solve a linear 1st order ODE. For example, the total derivative of $F(x,\ y(x)) = 2xy$ is $2y + 2xy'$, while the solution to $2y + 2xy' = 0$ is $2xy = C$. A conceptually interesting observation is that the RHS of the ODE can be nonzero during this process. This strengthens the analogy to the linear ODE, but unfortunately does not seem helpful for solving inexact ODEs, since isolating a differential form on the RHS that can actually be integrated ($f(x)\ dx$ or $g(y)\ dy$) never alters the scalar curl of $\begin{bmatrix}M(x,\ y) \\ N(x,\ y)\end{bmatrix}$ for any $M(x,\ y)\ dx + N(x,\ y)\ dy = 0$.
On the other hand, the process of solving an exact 1st order ODE can be viewed as the inverse of finding the work along a curve in a vector field. The routine MV Calc line integral exercise supplies a vector field and a curve, and the solution is work, whereas the ODE supplies the vector field, $\begin{bmatrix}M(x,\ y) \\ N(x,\ y)\end{bmatrix}$, and the work, typically $0$, and the particular solution is the curve (the general solution is the set of all possible curves that would have yielded the same work in $\begin{bmatrix}M(x,\ y) \\ N(x,\ y)\end{bmatrix}$). I've slightly conflated work at a point and work summed over an entire curve; technically $M(x,\ y)\ dx + N(x,\ y)\ dy = 0$ implies that the integrand of the line integral must be $0$, which in turn implies the weaker conclusion that the line integral evaluates to $0$.
Given all of this, I would expect there to be a strong conceptual link between the multivariable chain rule and line integrals over vector fields. Indeed, they should be more or less the same operation, perhaps to the extent that every multivariable chain rule problem has a dual work problem, and/or every work problem has a dual multivariable chain rule problem. However, this conclusion is prima facie strange, since the multivariable chain rule belongs to the domain of differentiation, while work belongs to the domain of integration.
Is there a way to see this conceptual link intuitively and directly, without considering exact ODEs as an intermediate step? How can it make sense to have such a dualism between the domains of differentiation and integration, which are generally themselves inverses? Is there a way to translate between work problems and multivariable chain rule problems for ease of computation?