You are missing the fact that if you actually exchange the integration and limit, you get $$\int_\alpha^\beta \lim_{\eta \to 0+}L_\eta(x-a)\, dx = 0 \quad \forall a \in \Bbb R$$
not the three-fold set of values obtained from integrating first, then taking the limit.
If you were to actually define a function by (1.7), we get, as pointed out by the first line of (1.8), $\forall x\ne a, \delta(x-a) = 0$. By the actual definition of integration (which much be considered an improper integral when $a\in [\alpha, \beta]$, since the integrand is not defined at $x=a$), this is sufficient to show that the integral in (1.8) is $0$ for every $a$.
The Dirac delta function is not a function as mathematicians normally define them, nor are integrals involving the delta function integration as mathematicians normally define it. Dirac introduced it as something of a play on notation - pretending to exhange the order of limit and integral at a place where such an exchange could not be justified. It was done as a sort of notational convenience so that the point model of a particle could be treated with the same mathematical formulas as continuously distributed masses.
Since that time, mathematicians have found many different ways of justifying the concept, including as an actual function taking on a new type of infinity at $x = a$, with an attendent extension of the definition of integration. But this is a far more difficult concept that just "$\delta$ is this limit".
Pnyisicists prefer to ignore such issues, assuming that as long as it gives good results, it must be justifiable, even if they no clue what that justification may be. Overall, this is generally true, but mathematicians have a greater awareness of how something that you thought was working perfectly can suddenly fail spectacularly, producing garbage (the most famous example being Russell's paradox, which forced a major rethinking of Set theory).
