Of course it does, the Agnelli family are intrinsically tied to the city of Turin and Juventus, they know the people and understand the club. That goes for anyone else foreign or Italian in comparison. I don't see how being a random fan compares to being an owner but it depends how proud you are perhaps.
That also doesn't mean to say a rich arab wouldn't do a good job. Considering to most football fans now throwing a (seemingly) blank cheque at everything is doing a good job, as long as they can watch the best players and boast to other fans. But then if everyone does that only one club can still win each title each year. If people think these guys genuinely care about the club they are kidding themselves, like anything they will become fond and attached, but they care about their investment and prestige. This is just boasting, comparing prestigious and monetary penises.