In finite dimensional vector spaces, orthonormality is defined as $\langle x_i|x_j \rangle=\delta_{ij}$ and the completeness relation is given simply by
$$I = \sum_i |x_i\rangle\langle x_i|.$$

To me, the most obvious extension of this finite dimensional case to infinite dimensions is to keep orthonormality as $\langle x|y \rangle=\delta_{xy}$, while the completeness relation is
$$I = \int |x\rangle\langle x|.\qquad(\ast)$$

Now in quantum mechanics, orthonormality of basis vectors $|x\rangle$ and $|y\rangle$ in an infinite dimensional vector space is **usually** defined as $\langle x|y \rangle=\delta(x-y)$, while the completeness relation is given by
$$I = \int |x\rangle\langle x| dx.$$

It seems to me, however, that my extension of orthonormality and completeness to infinite dimensions is equivalent to the usual extension.

As a demonstration that my version of completeness is consistent with my definition of orthonormality, let $| f\rangle$ be some arbitrary linear combination (integral) of basis vectors $|x\rangle$:
$$|f\rangle = \int f(x) |x\rangle dx.$$
Applying my version of the completeness relation to it:
\begin{align}
\int |x\rangle\langle x|f\rangle &= \int |x\rangle\langle x|\left(\int f(y) |y\rangle dy\right)\\
&= \int\int f(y)|x\rangle\langle x|y\rangle dy\\
&= \int\int f(y)|x\rangle\delta_{xy} dy\\
&= \int f(y)|y\rangle dy\\
&= |f\rangle
\end{align}
so equation $(\ast)$ does work as the completeness relation.

Is there anything wrong with defining things this way? If not, then why is the usual definition of orthonormality and completeness preferred?