If that were true, there'd be a riot every time a very famous person goes outside for any reason.
I'm sure she'd be approached and photographed and her privacy violated as much as people can get to her in a private lounge, but unless they were to advertise she is going to a certain airport at a specific time, it's incredibly unlikely she'd be mobbed. Ironically, flying publicly would make her movements harder to follow.
She can certainly afford to pay for 10 extra first class tickets for her staff, it'd most likely be much cheaper than owning her own jet. I'm sure the airports would also be thrilled to offer a private entrance and area for her/other famous people to be able to avoid even walking to her VIP lounge. Maybe they could help subsidize the airports instead of average people's taxes paying for their private airports in part.