How are you going to write generic functions that take functions as arguments and re-throw the errors thrown by these functions, if you use checked exceptions? Will you require that the acceptable functions only throw exceptions that you like? -- Then your generic function is close to being worthless...
If exceptions are encoded in function's interface, then they have to be in ABI, but then you must have non-trivial types available when marshalling data between two components, so, you cannot serialize the communication using some protocol with a fixed number of types (eg. JSON and friends), because now you need to account for the infinite variety of exception types.
Because of at least these two things, what I saw happen a lot of Java / C++ projects (god blessed me with very little C# exposure) was that as soon as a developer encountered a function with checked exceptions, a wrapper was written which changed the type into a runtime exception. This is so because exceptions are supposed to be handled separately, and often the author of the function has no idea how they need to be handled -- so they want to concentrate on the main goal of the function. Once functions grow into garlands of try-catch-catch-catch-...catch the focus is lost. It becomes very hard to understand why the function was written in the first place, because the error handling takes over every other concern.