It’s just a fashion trick to bring attention to the tie and add some depth to the look.
I also do it if I tie the tie a bit too long and I don’t have time to redo it. When standing a tie should end between your belly button and the top of your belt. A small window but that’s where it should end and sometimes, in a rush, you can mess that up. When that happens to me, I puff the tie with a tie bar and keep it moving.
I saw that you said that blue suit isn't possible in September in NYC. I'm curious as to why? (I'm just trying to learn because I know women's fashion and nothing about men's.)
My comment, if I recall, is that suit wouldn’t be possible until around Sept and that’s because the sports jacket is a heavy wool. That’s a very hot fabric and during the spring and summer, that’s torture in NYC. The humidity is insane. Not a good idea. Looking good is meaningless if you’re supremely uncomfortable.