You do lose the ability to use some features, that's true. Mostly RAII around the interface. You can still leverage it internally in the implementation, and if using context objects it would be even easier. The main pain point is if you want to let client of the library use their own allocators. It's still doable, but quite a pain.

Classes can be wrapped with a bit of effort. You do need to write the constructors and the destructors manually and invoke a pair of new/delete on the C side, but it's just as you would handle a type allocated by a C library itself. You'd use it the same way. You just have the liberty to have the implementation use (mostly) C++.

You can still use them inside your code, just not as arguments or return values from the API.

And it's not like that's a point in favor of C, since C doesn't have vector or unique_ptr either