You seem to ask why the specific contribution by Cauchy to the foundations of analysis in terms of his limit concept has not had major applications in physics and other fields (as have many of his other contributions, to complex analysis, combinatorics, elasticity theory, differential geometry, etc). There are two parts to this question:
(1) did Cauchy really make such a contribution, namely a strictly arithmetical definition of limit?
(2) Why doesn't a strictly arithmetical definition of limit have more applications?
I think the answer to (1) is clearly negative; Cauchy never gave a strictly arithmetical definition of limit. He repeatedly defined limit in kinetic or kinematic language, in terms of what appears to be a primitive notion in Cauchy namely that of a variable quantity, akin to Leibniz. What you take to be Cauchy's strictly arithmetical definition is actually Weierstrass's, in terms of epsilon-delta and alternating quantifiers.
Specifically, the notion of Cauchy sequence, while its mathematical equivalent is found in Cauchy's work, was actually defined by Cauchy in the language of infinite indices and the property of the corresponding terms in the sequence being infinitely close, or more precisely partial sums being infinitesimal. This makes Cauchy's procedures closer to their proxies in Robinson's framework than the Weierstrassian framework.
Cauchy on occasion exploited arguments that seem to go in the direction of our epsilon-delta proofs, but in none of them did he give an explicit formula for delta in terms of epsilon, a tell-tale of a modern epsilon-delta proof.
As far as the apparent lack of scientific applications of Weierstass's purely arithmetic notion of limit, it must be attributed to the fact that this tremendous accomplishment by Weierstrass had mainly applications in establishing firmer foundations for some procedures in analysis, especially study of Fourier series, where the 19th century literature was plagued by errors. In other words, this accomplishment is mainly significant for mathematicians rather than physicists and other scientists.