In my opinion, the significant difference in the mentality of teams in such competitions is that less titled teams have increased motivation, and more titled ones have decreased motivation. That is why the favorites here rarely reach the final (and by the way, they don’t worry much about it since the national championship and the Champions League are their priority) and the prize goes to the underdogs.
In the Brighton - Roma pair, Roma is the more titled team, so theoretically Brighton will have higher motivation. In addition, I would say that Brighton is stronger than Roma if we compare these teams based on results in the national championships, adjusted for the difficulty of the national championship.
I would say that is not exactly the situation, but some may not show the same passion for sure. Look at Liverpool, they had a 25 year old average on the caraboa cup, everyone talks about how "young" Liverpool was, on average Chelsea players were younger to be fair but that's fine.
If Liverpool doesn't care and do not play their best players then they may lose this one out too, that is just fine and could very well happen. I would say that it is not going to really be easy for Leverkusen to lose though, they are about to win the league, and the cup is a possibility so if they end up with Europa as well, that would be insane for them. I feel like they have a bigger shot at this than Liverpool if they care about it a lot more.
Liverpool is a different story, they historically try to win all the cups (even small ones) in which they participate. And by the way, given the situation due to Klopp’s upcoming departure, Liverpool could theoretically have increased motivation to win the Europa League since this is the only tournament that Klopp did not win with Liverpool.
Yes, Bayer have a good margin of safety and they can slightly shift their priorities towards the Europa League, but when your opponent is Bayern (even though they are strong only on paper), and you have never won the Bundesliga title, this is quite difficult to do.