Even if we can't rewrite everything up front, it's used to think about the problem at hand from first principles, ignoring the constraints of any particular language. From the vantage point "double dispatch" will never arise.
It's apparently not useful in languages with sum types, and it doesn't seem useful in an object-oriented language either (where you'd use the normal visitor pattern). So, when is it useful? The author doesn't have a practical example.
I guess there might be a language somewhere that has neither sum types nor methods? It's kind of a reach.
But, maybe it's useful educationally, as a good way to think about problems? That doesn't seem particularly likely either, considering that you can apply the visitor pattern just fine without knowing about Church encodings at all.
Disparaging "double dispatch" as "epicycles" while promoting Church encoding as "legit" just seems out of touch with how practical programmers think. I can't think of any reason why I'd want to use Church encoding to explain what's going on to a newbie.
Maybe there is some other situation where it's useful, but it's not explained.
"This post also explains how you can usefully employ the visitor pattern / Church encoding / Böhm-Berarducci encoding to expand your programming toolbox."
I said it was useful theory. E.g. a way to shrink a programming language to remove data types so there's less drudgery in proofs about that language. That's not a regular programming task in the slightest.
For example, using a different language means that the build system needs to support the other language, which can significantly complicate the difficulty for users of a library compared to a "pure" version written in the same language as the rest of the ecosystem. It may also limit portability, if the original language and the other language you pick don't support the same platforms. This isn't a good thing to inflict on users of your library.
To be a bit more concrete about it, suppose you are writing code in Rust to target WebAssembly. Pure Rust libraries are more likely to work than those written in a different language.
There are other ways to avoid this. In Go, you can use the "go generate" command to generate Go code from code written in another language, and check in the Go code, so downstream users of your library don't need any special tools. That's probably okay for private dependency on code written in another language, but less useful for datatypes used in public API's.
Someone talking about OO "epicycles" is ignoring most of the constraints that library maintainers face. It seems unsympathetic and out of touch.