A church encoded boolean is precisely isomorphic to every language's standard booleans (modulo strictness, perhaps) and doesn't offer any benefits; you're still forking the program based on the information content of a single bit.

Let's take the following function invocation, which can be expressed with Boolean literals or Church encoded booleans, I don't care:

  match true false

If you want to determine the significance of the boolean values passed to this function, it does not suffice to go to the definition of 'true' or the definition of 'false'.

Now take something like this:

  match caseInsensitive contains

Even though I have used descriptive names here, it's almost beside the point; I could just have easily have used nonsense names:

  match foobar quux

If you want to know what 'foobar' means, you can go to its definition, and see how it preprocesses a string and a pattern. You don't have to guess about the meaning of a bit.

As a result, the semantics of 'match' and its parameters are all communicated more clearly, with less room for error, and much more generality.

There need not be any syntactic overhead: it is merely the replacement of some flag with a lambda which cleanly encapsulates the effect that would otherwise be encoded in the flag. The way you invoke the function is the same, but instead of twiddling bits to get what you want, you pass functions whose meaning does not require (as much) subjective and possibly error-prone interpretation.

Note this also objectively simplifies the functions themselves, because they formerly contained conditional logic, but once you rip that out and give them no choice (invert the control!), they have less room to err, which makes them easier to get right, easier to maintain, and easier to test.

There is also another way to view the issue: with booleans, we first encode our intentions into a data structure (at the caller site), and then we decode the data structure into intentions (at the callee site).

Well, why are we packing and unpacking our intentions into data structures? Why not just pass them through?

Indeed, we do that by pulling out the code and propagating it to the caller site (possibly with names so you don't need significantly different syntax and can benefit from reuse). Then our code more directly reflects our intentions, because we're not serializing them into and out of bits.

I think the general principle applies to more than booleans, but it's easiest to see with booleans.