I'v avoided this thread for awhile, but I hope I'm able to jump in with a first person view, since I've been aware of this "rule," and the (BG) Billy Graham rules that surround it.
First, BG's approach of not criticizing his colleagues in ministry is one of his greatest legacies, alongside the financial accountability that is so crucial -- churches and ministries are more transparent and healthy because of this.
As for the rule about interactions with the opposite sex -- my DH and I use this rule, kinda like a Pirate Code (meaning it's a more a guideline, in a wink to the movies). It is not just for protecting fidelity, but also for safety from false accusation and, well, just safety. These days, it sometimes has nothing to do with affairs or the just the opposite gender. In my years teaching at a Christian college, I taught private music lessons, but my studio was in a well traveled corridor with a window in the door. On one occasion I was actually physically threatened by an male student angry over his grade. I was grateful that a male colleague heard the kerfuffle , peered in and intervened.
DH spent 20 years in pastoral ministry, and is now a college professor. If he meets with a female colleague, congregant or student, he might make simple adjustments. For example, he might meet in the open area of the church, at a comfortable table in full sight, but with enough space for the conversation not to be directly overheard. There have been times when he has just texted me when he's going to visit with someone and he might be alone -- we also use the "Find My Friends" feature on the iPhone, which means I could know, if I wanted to, exactly when he enters and leaves a location. Of course, it's really a way to avoid the phone calls, "Are you still at Target?" since he can just look and see now, lol!
FWIW, my view alongside DH, as he entered the private parts of people's lives (sometimes taking me along, when it seemed to make sense), I saw relationships torn apart by secrets and lack of care --- A marriage is strengthened first by the energy invested in each other, without which, boundaries are pretty meaningless.
As for the patronizing quality of the BG rule -- I've been on the other side of that, too. When I was a teen, the whole "pastors should only give side hug thing" became popular. I told my pastors, whom I had been close to since childhood, that if they were going to give me such a lame hug, they shouldn't bother! I also speak up regularly whenever someone asks me, (and yes, it's happened), to speak to a young girl about her "dress." I challenged one male elder to go to the junior department at Kohls and see if he could put together an outfit that was cute -- and to the knee. Gamely, he took me up on it, and came away with a new sense of how difficult it can be to run the gauntlet between the expectations of others. Finally, I did suggest to one elder, who questioned my willingness to kiss men onstage in opera (in gooey makeup and sweating under bright lights in front of loads of people), that I wasn't a Jezebel, but I might be a Deborah, or, better yet, a Jael! Truthfully, I don't think he knew who Jael was! https://en.wikipedia.org/wiki/Jael
I do not know VP Pence personally, but I now live in his state. My work colleagues have lobbied him in the Governor's office, and two friends at church worked with him on his staff. There are many who disagree with his policies (I don't often agree with him, either), but to a person they affirm that he was accessible, and willing to hear someone out. They all loved Karen Pence for her advocacy and simple ways -- which are getting her pretty much chewed up in D.C. I think it's appropriate to make sure that women who work in government have equal access to the VP, and that his "rule" doesn't affect his job. Heck, I'd offer to bring alone a "3rd party!" Rather than have one of Pence's aids in the meeting, I'd bring alone my daughter, or another young woman, inviting her into the halls of power with me -- to do a little "team building," you might say.