I found another reference to this "fact" on nlab which might need to be reworded: ncatlab.org/nlab/show/real+number. "This construction is equivalent to the construction by Dedekind cuts, at least assuming weak countable choice (which also follows from excluded middle)."

@JoelDavidHamkins Is Martin’s conjecture a formalization of this? In particular, does Martin’s conjecture imply that Hilbert’s 10th problem for rationals can’t be an intermediate degree?

But we can also show that Kolmogorov complexity solves the halting problem, and the halting problem isn’t computable, so Kolmogorov complexity isn’t computable. So transitivity it has a proof (not the standard one) by diagonalization.

I think to compute the area, we need not only that the boundary is zero, but also that we know how to enumerate all of the interior of the Mandelbrot set, which at least a decade ago we didn't know how to do without assuming the interior is only made of hyperbolic components.

In short, if two conjectures are true about the Mandelbrot set, namely (1) the interior is the union of the hyperbolic components and (2) the boundary has zero area, then yes, we know how to compute it in theory (although the calculations are hard). We also have an exact formula which can't be computed directly, and good estimates from Monte Carlo methods. But without assumptions (1) and (2), we don't know these methods will work, and we don't even know if the area is a computable real number as explained in the question above.