I was really hoping that someone would say, "There's an unwritten gymnastics rule that you don't show too much affection to a former student because it shows disrespect to the new coach." or something like that. It sounds like the concensus is that it's just bad behavior (like I suspected). Thanks, everyone, for your insight.
I've said it before (and was BLASTED for it), but people need to be honest with themselves in this sport. Some gymmies will never achieve high podium results because they lack talent. Some don't have the motivation. Some don't want gym to be that big of a part of their lives. They should embrace who they are and enjoy what gym can bring to them rather than focussing on placement. Focusing on what you don't have the capacity to achieve causes drama (usually from parents- myself included, though I think I've gotten better).
The same goes for gym owners. Some gyms simply don't have the technical level of coaching to take gymnasts to championship levels or even to become optional gymnasts. The gym that this gymnast came from is one of those. I think the HC hasn't realized what her gym is (or should be). A gym can be successful by focussing on its rec program and developing students to the point where they are ready for more technical coaching (i.e. have a team that focusses on getting it's gymnasts to Level 4 or 5) and then passing them on to coaches with the skills they lack. This particular gym is very fun for the kids, but if a gymnast wants to be competetive, it is simply not the place to be. If this coach were honest with herself, she'd see that she helped mold a prodigious talent and has passed her development to a coach that can take her to the next level before that coach will eventually wave to pass her on, too. There should be great joy in that.
As a volleyball coach, nothing makes me happier than seeing one of my kids move onto a travelling club team. I taught them what I could and now they're ready for the next step. I wish this coach would think like that. It's heartbreaking to see a child shunned.
(For the record, the gymnast is not my child, but is one of my daughter's friends.)