You have the history backwards. Differential forms came first; and the general integrability theorem actually preceded differential forms, going back to Clairaut, 1739-1740.
For an equation of the form $A dx + B dy = dC$, Clairaut used Taylor expansions to prove the necessity of $∂A/∂y = ∂B/∂x$ and indefinite integrals to prove sufficiency. Cauchy used definite integrals (1823): $C(x,y) = \int_0^x A(X,y) dX + \int_0^y B(0,Y) dY + C(0,0)$ for sufficiency. It might be possible to use the 2nd order Taylor's Theorem with remainder to directly establish necessity, eliminating the need to use any infinite series expansion. This requires only $C^2$-ness, which is what is already required for the theorem.
The exterior algebra (that is: where differentials anti-commute with each other) is from Grassmann in the 1840's. Before Maxwell stripped down Hamilton's quaternions to a vector algebra and applied it in his treatise in the 1870's, he used differential forms in his papers in the 1850's and 1860's; and also made much more use of them in the treatise than he did vectors. But he only made sparing use of Grassmann's anti-commuting rule, just one place in the treatise, as far as I know.
Here: this will clear things up. This is what you were really asking and looking for. https://sites.math.washington.edu/~morrow/335_17/diff%20forms%20history%20katz.pdf