I'm not arguing universally against what you're proposing by any means, but it doesn't seem like a clearly superior approach to me. The bottom line is that "let it break" takes many forms. There's your version, there's my type coherence version, there's Erlang's version with supervisor trees, etc. I think it's hard to argue that any of these is universally superior to the others.
This is far from the whole software industry, but it's a segment that tends to be of disproportionate interest to HN readers, given the startup-focused origins of this site.
This works fine in Haskell, thanks in part to laziness. There needn't be a tradeoff here.
In a strict language, a type error would mean the bricked off expression that doesn't type-check would spread like an infection to any expression that incorporated it.
In Haskell, I can dance around sub-expressions that don't type-check at runtime and evaluate things optionally.
The technique doesn't work in practice otherwise.
I don't see why. Functions in a strict language are evaluated lazily. Any type error within a function wouldn't spread beyond that function.
Even strict-by-default languages that let you easily make values explicitly lazy would mean type level churn, if not term level.
The value part is important because it means you can't really talk about partially implemented hypothetical results where you know how to handle one branch of a function, but not another. This is a super common divide-and-conquer technique in Haskell.
Could possibly half-ass it if you have some kind of panic functionality, but it'll fail unconditionally in a strict language when the result of the function is forced. In Haskell, if you don't use the bottom, it doesn't blow up.