To my mind this casts doubt on the importance of MDL.

I think its uncomputability already does that. When you make a computable version by limiting attention to some framework of descriptive capabilities smaller than universal computation, different choices of that framework will give you different measures of simplicity. What is simple in one framework may seem elaborate and baroque in another. Or as some military strategist once put it:

"To the foot-soldier, the strategy of a general may seem obscure, shrouded in shadows and fog, but to the general himself, his way is as plain as if he were marching his army down a broad, straight highway."