tldr No. The world would be like Star Trek.

I don't think so. A fair bit of the trust fund kids that I've known are pretty unhappy. If I had to guess, they're unhappy because society uses wealth as a proxy for worth. While their wealth and perceived worth are high, it has nothing to do with them.

Humans measure things against other things, and we measure ourselves against other people. Trust fund kids have fewer examples to measure themselves against, so unless they carefully select samples, they probably have a hard time gauging their worth.

In a post-scarcity society, post-scarcity is common and we'll cease measuring our worth using wealth as a proxy. I'd guess we'll measure worth by politics, leadership, charisma, charm, intelligence, and physical attractiveness. If you found a properly isolated clade of trust fund kids, I'd guess they measure worth on dimensions like those.

Haha. Remember that one person's utopia can easily be the dystopia of entire nations of other people.

Feeling purposeless and without a "place to go, things to do" can hit lots more people than trust fund babies. Retired people without a good idea of what they want to do post-retirement can also feel at a loss, and without value to society.

I'd love the outcome to point towards a future as described in Ian M Banks' Culture series which would be more trust fund baby-esque.
However, I'm afraid the future as described by James SA Corey's The Expanse series is more likely. In particular the population on Earth and their reliance on "Basic Assistance."

I wouldn't say good, but it might be accurate. Virtually everyone who doesn't have to work shares the problem of purpose and respectability.