Why do we restrict the definition of Lebesgue Integrability?

Technically speaking, the function $\displaystyle{f(x) = \frac{\sin x}{x}}$ is not Riemann integrable on $(0, \infty)$, but rather improperly Riemann integrable on $(0, \infty)$.

The construction of the Riemann integral only works for bounded intervals. We can extend this construction to unbounded intervals like $(0, \infty)$, but that requires an additional limiting process. It is the first construction (Riemann integrals for bounded integrals) that the Lebesgue integral generalizes.

You could just as well ask the opposite question: why do we define Riemann integration in such a way that an integral can be convergent without being absolutely convergent? The definition of each type of integral "is what it is," and the way the Lebesgue definition is defined there is no need for improper integrals like in Riemann integration.

We could simulate an improper integral with Lebesgue integration by taking a limit of Lebesgue integrals over bounded regions. But that's not something that's usually of interest in the Lebesgue theory.

The things that are of interest are convergence theorems like the dominated convergence theorem.

Dominated convergence theorem: If $(f_n)$ is sequence of measurable functions, $|f_n|<|g|$ for $n \in \mathbb{N}$, $\int |g| < \infty$, and $f_n\to f$ pointwise then $\int f < \infty$ and $\int f_n \to \int f$.

In that theorem, the dominating function $g$ needs to be absolutely integrable. Your example, in fact, can be modified to give a counterexample to this statement:

False: if $(f_n)$ is sequence of measurable functions, $|f_n|<|g|$ for $n \in \mathbb{N}$, $\int g < \infty$ when the integral is computed in the improper sense, and $f_n\to f$ pointwise then $\int f < \infty$ (again in the improper sense) and $\int f_n \to \int f$.

The actual theorem requires $\int |g|$ to be finite. Since that is the sort of condition that we work with most of the time, we use the word "integrable" for it to save space. We can still recapture improper integrals if we have to, but they're rarely of interest in the context of Lebesgue integration, so we don't want to spend a good word like "integrable" on them.

You might be interested in Henstock-Kurzweil (HK) integral. Its definition is an easy modification of Riemann integral. All Lebesgue-integrable functions are integrable in the HK-sense, and so is your function $\sin(x)/x$. The usual theorems from Lebesgue theory (such as the dominated convergence theorem) have extensions to the HK theory. On top of that, Newton-Leibniz formula holds for any function admitting an anti-derivative (which is not true in Lebesgue theory).

That being said, you definitely need Lebesgue theory for spaces different from $\mathbb{R}^n$. Notice also that $\mathbb{R}^2$ is isomorphic to $\mathbb{R}$ as a measure space (and that any sensible measure space is isomorphic to an interval); you would lose this unity if you wanted e.g. $\sin(x)/x$ to be integrable.