Constructivist basically means being able to be explicit. Dedekind cuts and Cauchy sequences are not necessarily constructivist though something described by one of them can be explicitly descriptive for some applications. Any approach which produces all real numbers as commonly accepted will fail to be explicit in all cases as such explicitness presumably implies the real number has been expressed uniquely with finite strings and finite alphabets which can describe at most a countable number of them.

The decimal numbers, for example, can be viewed as an infinite converging sum of powers of ten. Theoretically one could produce a description, but only a countable number of those could be written down in finite terms (some kind of finite recipe). So those finite ones could fall in a constructivist camp, but the ones requiring an infinite string to describe would, as far as I understand constructivism, not fall under being constructivist. To be clear, the finite string doesn't have other be explicit about how to produce the numbers, just that it is naming the thing and it can be derived from that. So square root of 2 names a real number and there is a process to compute out the decimals so that exists in a constructivist sense. But "most" real numbers could not be named.