This scenario basically happened to Tyson Jost in the NHL last year, the difference being he was traded rather than cut. He’s on record confirming exactly this, he was happy for his old teammates but he bounced off of social for a while because it was actually making him sick.

Honestly feel bad for him, he was popular among fans and I know some of us wanted to see the team add his name on the Cup too.

ETA the quote - from an article by The Athletic so avoiding the paywall:

“Mentally, I was already dealing with some stuff and then you see that,” Jost, 24, said, staring at the wall in front of him. “I mean, I guess you could kind of imagine how it is. You dream of winning the Cup and then you see your best friends doing that. It was just … a lot of tears. Just sucked. I’m sure in the long run it’s going to make me a better person and player, and you just got to deal with that stuff. It’s part of the business. I know I’m not the only one this has happened to (in NHL history), but it definitely did suck.”