Some thoughts on this:
1. Alex is, at least in my opinion, entitled to take a weekend for himself and go riding, which he loves. JJ has a lot of competitions, and it really isn't reasonable to expect Alex will be at every one of them.
2. As Alex noted, JJ needs to show that he isn't obsessing and dependent on Alex.
3. JJ had to leave earlier than Alex could have, so the scenario would have necessitated two private jets...one for JJ, and one for Alex. Not that money is a huge issue, but that's a bit expensive.
JJ demands and expects a private jet no matter where he goes. I can just see his bulging eyes and his total temper tantrum when someone suggests he should fly on a (gasp) airline! All of these guys have their idiosyncrasies, but JJ has been consistently adamant that he wants the easiest, most painless, and most luxurious modes of transportation.