We used to have cross links at iGi Partners when it was first introduced in the constitution (v3.0 if I remember well). The concept was very attractive and fun to play with so I must confess I used it abusively by creating intellectual cross links (at that time you couldn't test governance proposals by asking “do you have a concrete situation of your tension?”). For instance, we had a GCC with many sub-circles including a Marketing Circle and in the GCC there was a “Graphic Design” role and I proposed to create a cross link from the GCC to the Marketing Circle just because I thought it would be great for me to show up in that role in the Marketing Circle. I think it was useful indeed because I could talk from that role, give my perspectives in tactical meetings and I could also take projects asked by Marketing roles. So it helped but only because I think it was at an early stage of our Holacracy practice and I was bringing almost all my tensions to tactical meetings and not processing them outside of it. When we noticed that cross link roles were less and less used in the target circles meetings, we removed all of them. It didn't hurt, made sense at that time and was a nice experience. Maybe cross links were the sign that the circles's membranes were not well defined and needed to evolve because there were lots of interaction between several roles. A cross link is, as you are aware of, kind of a whole created between two circles' membranes, another link channel to process tensions so it might be a good enlightener that the structure is not the requisite one and maybe using cross links is an inevitable way to evolve the structure later.
Another case that occurred where we used cross links is when we were working on our Holacracy Comic Book. This huge project was gathering multiple roles across multiple circles so we organized specific meetings (neither tactical nor governance), out of any circle. We worked like that for a year and some months before the launch of the Comic Book, we needed more synchronization and we felt tensions to evolve roles, set expectations, etc. so we created a circle call Holacracy Comic Book (sub-circle attached to the GCC) where we only had a Lead Link and Cross Links from all the role implied in the project (no Rep Link as a Circle with only a Lead Link and Cross Links has no Rep Link per constitution). It was very useful, productive and was the right thing to do to help that project launch because of the energy it required. Once the project was launched, the circle was removed and the cross links as well.
Another case you might be aware of is the Ecosystem Circle within HolacracyOne using Organization Cross Links to manage a relationship between two companies that have an agreement and are working closely together. We have a Cross Link from iGi Partners representing the whole company within the Ecosystem Circle. Any accountability can be added on it and it allows us to process tensions and drive the agreement with governance dynamically. We have some clients exploring that as well as they have kind of a franchise/licensing agreement as well.
I have lot more of experiences were it was good or at least a good start to process tensions but not the perfect way to do it (inevitable way maybe until the structure evolves?).
Going back to what you said “Since launching Holacracy, we have avoided the use of the Cross Link Role/Policy. In cases where circles felt the need to be better connected/aligned with other Circles, they have created liaison/advisor roles. This has seemed to work well so far.”
I don't know the tension behind that solution but I think one way to achieve alignment/connection could be done by putting accountabilities on the circles themselves to set expectations clearly.
I hope that my experience with cross links can help you.