Yes, an airport limits the amount of people, has a very high coverage of surveillance and a high ratio of security staff as well as an entry barrier and dedicated VIP areas. A generic place outside has none of that. Although feel free to elaborate on how an airport is worse for security than just being on a street, anywhere.
To your second point, sure she doesn't need to own them like nobody else does, but the issue (for me) is not primarily that she (or anyone) owns one, but that they [private jets and private airports] exist, and they're subsidized by us as it was pointed out above. If anything, they should be priced outrageously so using them would come down last resort or emergency situations, and the money from that could help balance the cost of the "public" infrastructure. This is a failure of the government, but equally so of the rich who choose to continue using them for their luxury.