Escape analysis is tough in C, and data returned by pointer may be pessimistically assumed to have escaped, forcing exact memory accesses. OTOH on-stack struct is more likely to get fields optimized as if they were local variables. Plus x86 has special treatment for the stack, treating it almost like a register file.
Sure, there are libraries which have `init(&struct, sizeof(struct))`. This adds extra ABI fragility, and doesn't hide fields unless the lib maintains two versions of a struct. Some libraries that started with such ABI end up adding extra fields behind internal indirection instead of breaking the ABI. This is of course all solvable, and there's no hard limit for C there. But different concerns nudge users towards different solutions. Rust doesn't have a stable ABI, so the laziest good way is to return by value and hope the constructor gets inlined. In C the solution that is both accepted as a decent practice and also the laziest is to return malloced opaque struct.
Heap allocations, yes; pointer indirections no.
A structure is referenced by pointer no matter what. Remember that the stack is accessed via a stack pointer.
The performance cost is that there are no inline functions for a truly opaque type; everything goes through a function call. Indirect access through functions is the cost, which is worse than a mere pointer indirection.
An API has to be well-designed this regard; it has to anticipate the likely use cases that are going to be performance critical and avoid perpetrating a design in which the application has to make millions of API calls in an inner loop. Opaqueness is more abstract and so it puts designers on their toes to create good abstractions instead of "oh, the user has all the access to everything, so they have all the rope they need".
Opaque structures don't have to cost heap allocations either. An API can provide a way to ask "what is the size of this opaque type" and the client can then provide the memory, e.g. by using alloca on the stack. This is still future-proof against changes in the size, compared to a compile-time size taken from a "sizeof struct" in some header file. Another alternative is to have some worst-case size represented as a type. An example of this is the POSIX struct sockaddr_storage in the sockets API. Though the individual sockaddrs are not opaque, the concept of providing a non-opaque worst-case storage type for an opaque object would work fine.
There can be half-opaque types: part of the structure can be declared (e.g. via some struct type that is documened as "do not use in application code"). Inline functions use that for direct access to some common fields.