There are a lot of results in elementary number theory that can be proved with the quadratic reciprocity law. In such a proof you usually have to invert some Jacobi symbol $(a/b)$ and then reduce the numerator modulo the denominator. For number fields that are not Euclidean with respect to some simple map you have a problem if you want to follow this route (the same goes for applications of quadratic and higher residues to cryptography, although this is mostly a theoretical business). In principle, Dedekind-Hasse will also do the trick in some cases.
If the ring of integers you're interested in is not Euclidean for the canonical norm, the first idea is to modify it. You could give prime ideals a different weight (weighted norms), or allow division chains in which the norm does not necessarily get smaller in every step (k-stage Euclidean rings), or try some version of Dedekind-Hasse. But if (given a pair $(a,b)$ of elements in a ring) you want to make the norm of $ka-bq$ small, you need more than just the knowledge that a suitable $k$ exists: you need a method for finding $k$ (in addition to finding $q$), perhaps by showing that you can select it from a finite set of elements with bounded norm or something similar.
Edit. The Euclidean algorithm is closely related to continued fractions, and the latter are routinely used for doing calculations of units and ideal class groups of real quadratic number fields. For number fields that admit a Euclidean algorithm, something similar can be done: Hurwitz and Mathews worked out a theory of continued fractions over the Gaussian integers, and people like Arwin, Trinks, Degel, Lakein, Stein etc. generalized this to complex Euclidean number fields and used it for computing units and class numbers. I am not aware of too many recent contributions in this direction, but a short search has at least revealed D. Fried, Reduction theory over quadratic imaginary fields, T. Number Theory 2005.
