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$\int_a^b \left( \psi_2(x) - \psi_1(x) \right) dx < \epsilon$

I think using this definition is easy and geometrically intuitive, and on the other hand working with this definition prepares you conceptually for the Lebesgue integral where you juggle "simple" functions instead of step functions. Thus, you already get a nice piece of the Lebesgue point of view.

The definition also demonstrates the broad principle that to construct an object in real analysis which should be a real number, one often needs a good way to overestimate/underestimate the object you're going for, and there are plenty of examples of that in real analysis outside of integration (liminf and limsup being the simplest) -- after all, this is just one point of view on how the real numbers are constructed.

To give an example of the ease of use: to prove the fundamental theorem, suppose $F$ is continuous with a Riemann integrable derivative $F' = f$ and let $\psi_2$ be a step function above $f$ which induces the partition $a \leq x_1 \leq x_2 \leq \ldots \leq x_{n-1} \leq x_n = b$. Then the total change of $F$ from $a$ to $b$ is the sum of the "small" changes $F(b) - F(a) = \sum_{k=1}^n (F(x_n) - F(x_{n-1}))$ which is less than or equal to $\int \psi_2(x) dx$ by the mean value theorem. Similarly for the other inequality.

Thus, one can go pretty far with this definition, but the results which separate the Riemann integral from the Lebesgue integral (e.g. that $g(f)$ is Riemann integrable for $g$ continuous and $f$ Riemann integrable or the fact that Riemann sums converge to the Riemann integral), one needs to use the observation that step functions can't be close in terms of area without being uniformly close on all but a small set. You could think this feature is either a clarification or a disadvantage. One certain disadvantage is that it is not the right point of view for integrating vector-valued functions. So you might decide it's better to define the integral in terms of Riemann sums in the first place (giving more of a "metric space" point of view and less of an "ordered space" point of view). Or you might even decide to skip some of these other topics, depending on your point of view and time available.

One available compromise is to just work with the following definition of a Riemann integral (which works fine in ${\mathbb R}^n$ as well:
A bounded function $f$ on $[a,b]$ is Riemann integrable if and only if for every $\epsilon > 0$ there are step functions $\psi_1, \psi_2 : [a,b] \to {\mathbb R}$ such that
$\psi_1 \leq f \leq \psi_2$
$\int_a^b \psi_2(x) - \psi_1(x) dx < \epsilon$