Here's a direct solution in case you don't want to end up repeating the second half of Riemann-Lebesgue's proof.
You can use the partition $P$ for which $U(f,P)-L(f,P)<\epsilon$ to argue that for all partitions $Q$ with mesh$(Q) = \|Q\|< \delta_P$, the Riemann sum will be somewhere close to $U(f, P)$ and $L(f, P)$.
Assume that $P$ has $N$ points $\{p_1=a, p_2, \cdots, p_N=b\}$, partition $Q$ has $N'$ points $\{q_1=a, \cdots, q_{N'}=b\}$ and $|f| \leq M$ in $[a, b]$ (this should hold for some $M$ or Darboux integral won't be well-defined).
Now take $\delta_P< \min(\|p\|, \frac{\epsilon}{MN})$. Now for each $i \leq N'$, either $[q_i, q_{i+1}] \subset [p_j, p_{j+1}]$ (for some $j\leq N$), or $p_{j-1} \leq q_i \leq p_j \leq q_{i+1} \leq q_{j+1}$. The latter can happen at most $N$ times, so the area under $f$ for such cases can be at most $N \times M \times \delta_P \leq \epsilon$. For the rest, the sum happens to be nicely sandwiched by $U(f,P)$ and $L(f, P)$, proving the Riemann integrability.
Let $\mathcal{P}$ ($\mathcal{P}_U$) be the collection of all partitions (uniform partitions) of $[a,b]$.
For a bounded function $f$, we have $L(P,f) \leqslant U(Q,f)$ for lower and upper Darboux sums corresponding to arbitrary partitions $P$ and $Q$. It follows that
$$\sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}}U(P,f), \quad \sup_{P \in \mathcal{P}_U}L(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)$$
Since $\mathcal{P}_U \subset \mathcal{P}$ we have
$$\{U(P,f) \,|\, P \in \mathcal{P}_U\} \subset \{U(P,f) \,|\, P \in \mathcal{P}\}, \quad \{L(P,f) \,|\, P \in \mathcal{P}_U\} \subset \{L(P,f) \,|\, P \in \mathcal{P}\}$$
Hence,
$$\tag{*}\sup_{P \in \mathcal{P}_U}L(P,f) \leqslant \sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}}U(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)$$
Darboux integrability with respect to uniform partitions means that
$$\sup_{P \in \mathcal{P}_U}L(P,f) =\inf_{P \in \mathcal{P}_U}U(P,f),$$
which, in view of (*), implies that $f$ is Darboux integrable with respect to all partitions since
$$0 \leqslant \inf_{P \in \mathcal{P}}U(P,f)- \sup_{P \in \mathcal{P}}L(P,f) \leqslant \inf_{P \in \mathcal{P}_U}U(P,f)- \sup_{P \in \mathcal{P}_U}L(P,f) = 0,$$
and
$$ \inf_{P \in \mathcal{P}}U(P,f)= \sup_{P \in \mathcal{P}}L(P,f) $$
Best Answer
This can be done by using the characterization that the difference between upper and lower sums can be made arbitrarily small, which is one of the very first results when studying Riemann integrability.
The idea is to "manage the bad points", the bad points being where you have the jump discontinuities, since as you point out, the supremums and infimums kind of mess up the Riemann sums at these points.
Let $\epsilon>0$ be given, let $M=\max_{x,y\in[a,b]}|f(x)-f(y)|$, and since we have $n+1$ jumps, set $\delta=\frac{\epsilon}{M(n+1)}$. Now consider the partition $P'=\{x_0,x_0+\delta,x_1\pm\delta,x_2\pm\delta,...,x_{n-1}\pm\delta,x_n-\delta,x_n\}$. Can you bound $U(P',f)-L(P',f)$ by something small?