I have a little headcanon about Rufus & Dio (& Palmer):
I think Palmer would have tried to take Rufus out to celebrate birthdays when the President wasn't around, but it tended to be bars & casinos because Palmer isn't exactly a responsible adult. Who's going to tell a Director no, go home, that teen's clearly underage? Except Palmer often got distracted by escorts and wandered off. Whatever, being left alone at a booth near one of the card tables was still better than having to deal with his father, so Rufus didn't care TOO much.
One year when Dio was trying to get investment/do market research before opening the Saucer, he saw this kid sitting alone. He got Palmer to introduce them, then when Palmer wandered away again Dio squished himself into the booth opposite Rufus and spent the evening teaching him coin tricks.
They stayed in casual contact afterward, which is also why Rufus had Dio's number and could arrange that exhibition match so quickly.
This is so good...it's crazy you dropped this when I was just contemplating Dio's history yesterday (I read the Corel Region intels while playing Rebirth, which I'd sort of forgotten about in my mission to complete other tasks).
It makes sense because Dio is kinda warm and soft, especially to younger people (albeit obviously quite eccentric), but he'd have to know in the back of his mind that being kind to the President's heir would be good for future business. It also works for my undying love of Rufus being grandiose and theatrical in his own right. Of course, Shinra as a corporation is full of nutjobs and weirdos, but Dio is one of a kind, right? Bit of a 'the difference between a villain and a SUPER villain is PRESENTATION!' situation.
And then anytime Palmer ends up taking him to said casino, little adolescent Rufus is (secretly) craning his neck looking for that big weird guy again. He's entertaining, at the very least.
A bit tangentially, I love Dio as a world building character because it can be easy to collapse the inherent nuance of FF7 into "EVERYTHING SHINRA" even though that undermines the themes about good people being caught up in, throttled by, or doing malicious noncompliance within a corrupt system. Dio probably found Shinra to be distasteful in many ways, but he used a can-do attitude and business acumen to get as free of their control as possible. It isn't perfect, but sometimes it's the only way to find relative freedom in an oppressive, capitalist, outright threatening system. He has enough juice in present day to resist Shinra on his own or subvert their operations when he disagrees with them. Dio, Reeve, Mayor Domino, and even newer supporting characters like the Shinra Middle Manager illustrate the complexities of such a society. The moral grey area isn't only provided by people like the Turks or Avalanche. The techno fascist overlords force difficult choices out of everyone, every day.