I think this is key. RC pointers don’t give you the ability to say: ok, I now know that this subset of the graph is dead because reasons so delete it. In RC, you’d have to walk the graph to break cycles. That’s a rather important difference from how you’d manage a cyclic data structure in C, C++, Java, C#, or any GC’d language. In C/C++, you’d delete what you know to be dead; in Java et al it would be enough to kill the references that point at the outer edges of the dead subgraph. But you wouldn’t have to walk the subgraph to detangle the cycle.
I think it really is worse than that; ie as you say, folks will just do anything but RC in that case. In Rust the safe bet would be to just shove all the nodes into an array and use indices (yuck!!!). But likely a lot of code with use raw pointers instead (also yuck). The whole thing where you reference count cyclic data structures is just too painful for sensible people
That’s exactly what weak references do.
For a tree with parent pointers, I'd absolutely use Rc, with Weak for the parent pointers.