Instead of defining a smooth manifold to be a manifold whose gluing functions are smooth, what would happen if we defined it as an $n$-manifold $M$ which has an embedding into $\mathbb{R}^{n +1}$?
A smooth map between manifolds $e_M : M \hookrightarrow \mathbb{R}^{n+1}$ and $e_N : N \hookrightarrow \mathbb{R}^{n+1}$ would be a continuous function $f : M \to N$ along with a smooth function $g : \mathbb{R}^{n+1} \to \mathbb{R}^{n+1}$ such that $g \circ e_M = e_N \circ f$.
Would defining them this way be equivalent?
For a start, the Klein bottle would no longer be a smooth manifold, as it has no embedding in $\Bbb R^3$. Nor would any non-orientable closed $2$-manifold.