This is a real pain point and I run into the same tension in systems where data crosses serialization boundaries constantly. The prototype-stripping problem you're describing with JSON.parse/stringify is a specific case of a more general issue: rich domain objects don't survive wire transfer without a reconstitution step.
That said, I think the Temporal team made the right call here. Date-time logic is one of those domains where the "bag of data plus free functions" approach leads to subtle bugs because callers forget to pass the right context (calendar system, timezone) to the right function. Binding the operations to the object means the type system can enforce that a PlainDate never accidentally gets treated as a ZonedDateTime. date-fns is great but it can't give you that.
The serialization issue is solvable at the boundary. If you're using tRPC or similar, a thin transform layer that calls Temporal.Whatever.from() on the way in and .toString() on the way out is pretty minimal overhead. Same pattern people use with Decimal types or any value object that doesn't roundtrip through JSON natively. Annoying, sure, but the alternative is giving up the type safety that makes the API worth having in the first place.