There have been multiple posts regarding the connection between Darboux & Riemann integrability, but most of them are only proofs for $\mathbb{R}$. The proof generally uses the following idea:
Let $[a,b]\subseteq\mathbb{R}$. If a bounded function $f:[a,b]\to\mathbb{R}$ is Darboux integrable, then by the definition of supremum and infimum, for any $\varepsilon>0$, there exists some partition $P_0=\{x_0=a,x_1,x_2,\dots,x_{N-1}, x_{N}=b\}$ such that $$ U(f,P_0)<\inf_P U(f,P) + \frac{\varepsilon}{2} \quad\text{and}\quad L(f,P_0)>\sup_P L(f,P) - \frac{\varepsilon}{2} $$
Then, by choosing $\delta=\min\left\{x_1-x_0, x_2-x_1, \dots, x_N-x_{N-1}, \frac{\varepsilon}{2M(N-1)}\right\}$, we can show that $f$ is also Riemann integrable.
The $N-1$ in the denominator is to account for the (at most) $N-1$ intervals in the new partition $P'$ (where $|P'|<\delta$) which are not a subset of any interval in the original partition $P_0$. The $N-1$ was derived from the fact that there are $N-1$ points in $P_0$ that is strictly between $a$ and $b$ (i.e. $|P_0\cap(a,b)|=N-1$).
However, this method of proof does not translate well into higher dimension, $\mathbb{R}^n$. At $n\ge2$, we no longer have the luxury of just considering the finite $N-1$ points; instead, we will have to reckon with an infinite number of points. For $n=2$, this will be a line; for $n=3$, this will be a plane; for even higher dimension, it will be hyperplane but either way, infinite number of points.
As such, we can no longer choose a reasonable $\delta>0$. Any solution I can think of to fix this leads to some form of circular argument, whereby we need $\delta$ to defined $P'$ but we need number of intervals in $P'$ to define $\delta$. So how do I reconcile and salvage this proof to show that Darboux integrability implies Riemann integrability in $\mathbb{R}^n$?
P.S. Rather than providing a totally different way of approaching this proof (e.g. consider integral about each of the $n$ directions separately), I would appreciate if the answers point out how to salvage this discrepancy or why there is no way to reconcile this discrepancy.