Its a fair rule, and in my experience, fairly common. In smaller, private barns I would always ask before bringing out friends/family to see my horse (none were going to ride) purely as a courtesy. I want them to know what is going on at their own property, and its just smart to run it by them in case there is a schedule conflict. “oh, we are spraying for weeds and pulling a tree out at 10am on Saturday, maybe your guests can come Sunday?” stuff like that. It helps everyone have a more pleasant experience.
As far as leasing goes, there are have been many threads here discussing how everyone has their own definition of what leasing is. It might mean one thing to you, but another to the BO. This is why everyone is encouraging you to talk to the BO about the situation and just get on the same page.
Should most of this been discussed before moving your horse in? Absolutely. That is on BOTH you and the BO. Its something you should have asked when touring the barn “Hey I can’t get out the barn tues/thurs due to school/work, so I have my friend jane come ride Dobbin so he can get some exercise. She’s really experienced and this arrangement has been working for us for X amount of time. Is this OK? What forms will she have to sign?” or whatever the situation is.
Anecdotally, I have seen this work out well and also be disastrous. One the good side, its nice to have someone who can hop on your horse to get him out if you can’t make it to that barn that day, or a more experienced friend who might be able to help with a tune-up or two.
On the other, there was one boarder who would “lease” her horses out to ANYONE. I’m talking unsupervised rank beginners who would run around the arena tacked up wrong, then put horse away wet. It was dangerous for everyone. Eventually that boarder got kicked-out since she refused to screen or teach the people she would sub-lease her poor horses to. It can get messy real quick.