Those are some landmark hearbreaking moments for sure. I guess the only thing I can say is that they were good enough to be in position to have those heartbreaking moments. If they were sitting at home out of the playoffs, they wouldn't have had those big games.
It's interesting that you said 2014 was the saddest sports moment in your life. I'm not sure it was the saddest for me because at a certain point I just knew they were going to blow it, I'm trying to think of a worse one. I think I might have taken losing the Super Bowl in '97 harder, because as you say we were huge favorites (although I looked at the Denver team, and I thought it was a mistake that we were favored so heavily). The NFC had won 13 straight Super Bowls at the time, so we were the one to blow the streak. Also, the Broncos had never won a Super Bowl at that time, and we had never lost one, so both those streaks were done. Ron Wolf made his "fart in the wind" comment, and it really did feel like the end of something, and it was. We had the chance to be a small dynasty, and blew it. It was a good game though.