Depends on the situation the team is in. If a team expecting to win 7 or 8 games ends up in a high-ranking bowl game, or if a team expecting to win 1 or 2 games makes a bowl (like Northwestern), they certainly don't see it as a consolation. If a team has national title expectations and ends up outside the playoff, then yes, they see it as a consolation prize.
I was young, but I remember the pre-BCS era. I agree that it felt different. People did assign more meaning to bowls that weren't going to determine a national champion. My question is why. The dynamic that exists today also existed in the pre-CFP and even pre-BCS eras. There was a bowl(s) that determined a national champ, there were bowls that didn't, and I agree that people assigned more meaning to the bowls that didn't. Rightly or wrongly, there has been a cultural change that has affected how players, fans, etc. view these bowl games.
Personally, I just think in our sports culture today there is simply more of a desire/need to crown a single team as a national champion, and to see most other teams that didn't win a national championship as having failed or underachieved in some way. There's just more of a need to do that in sports today than 20 years ago, and definitely more than 40-50 years ago. I'm not an NBA fan, but the comments Giannis Antetokounmpo made after getting knocked out of the playoffs last year sums up how a lot of fans, media, perceive performances that don't result in a championship, and his answer was great.