I am not sure what your last statement means.

Constructible reals are continuous over [0, 1] in that there are no gaps between any interval between constructible reals [r1 r2] that are not filled by more constructible reals (and in fact, the cardinality of the constructible reals within any interval is the same, i.e. countably infinite, in a fractal way).

So there is no obvious motivation for anything but constructible reals from that standpoint.

Unconstructible reals were invented (or at least used) by the mathematician Cantor to explore ideas about different infinities. A "real" number with infinite decimal digits but not any finite description lets him create a number space larger than the constructible reals, a larger infinite cardinality.

So there is nothing missing in a [0 1] constructible interval. Or to put it another way, constructible numbers are closed. There is no sqrt(-1) situation requiring unconstrucible numbers to fill, like the square root of -1 required imaginary numbers (or geometric algebra dimensions) to fill.

But [0 1] contains the higher infinity of unconstructible reals in it, if you want. But I am unaware of any claim that they solve any problems by being included, other than exploring interesting puzzles related to unconstructible numbers as interesting ideas in themselves.