But, if, by "Star Trek future," you just mean a post-scarcity future, then, maybe? There would be no reason everyone should need to be a PhD, but also no reason why everyone couldn't be [1].
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[0]: https://memory-alpha.fandom.com/wiki/Risa
[1]: Everyone with the intellectual capacity, that is. One certainly wouldn't expect severely developmentally delayed individuals to possess graduate-level knowledge of any subject. Gradeflation only goes so far. :-P
In the "Star Trek" you don't see broad society, sometimes you see normal people, but series is mostly officers.
You don't want Phds only society because that is a lot of waste. From my point of view bachelors or engineering degree is mostly what will be needed in the future.
Because for me Phds are mostly on thought level of someone who has done bachelor. Staying in academia is a career choice and not that you are chosen to become Phd because you were super smarter than others. It also turns out that to be successful Phd it is more about politics at the university that "beeing ultra smart". True that you still have to be smart to play politics but don't have to point fingers at stupid politicians that handle that as well without having even masters degree.
I chose career in building web applications, this is what I am proficient at and I learned it on the job/on my own.
If I would choose to follow Ph.D. I would optimize different things just like your father. Well I believe your father is not proficient at building web applications and I don't think he could be as proficient as me in 6 months. The same for other things I could not become professional football player in 6 months or conduct research in proficient way.
In Star Trek's future, space travel is commonplace and there are almost certainly non-experts in starship crew roles; Ensign Redshirt, who gets killed by the planetside monster, probably doesn't have an advanced degree.
Maybe you're on to something....
Jokes aside, whenever people point to "Star Trek" as an example of post-scarcity society I recoil at just how over-simplified and not well thought out the comparison is.
If you run this trend forward even 50 years what jobs do you really think will exist that wont basically be scientifically or academically inclined?
Intersaller is probably more on the mark. Blight striking nearly every staple food is the real horror in the night. Cavendish bananas are under massive threat of a fungal disease wiping them out, forever. The same thing can happen at anytime to other overproduced mono-crops.
But then let's take humanity at it's current state. If everyone believes working with their hands is beneath them, are we really going to reach automation peak? Ignoring the scifi fantasy of skynet ai, you really think there wont be exploitation? The one thing humans are brilliant at, it's exploiting a system. You have to pacify a human to stop them from min-maxing. It's what we do. We are already seeing the adverse affects of too many systems online. Ransomware is not a problem, it's a goddamn well oiled industry. The future luddite wont just break a loom or two. They can stop fuel for tens of millions of people. Turn off cooling/heating to homes. And if wireless bots run farms... well, I dont think you need a good imagination to figure out what will happen there.
I do find this illuminating, though, because if someone fundamentally believes problems are caused by a lack of resources - or more specifically a lack of access to resources - then it makes perfect sense to them to think post-scarcity will be a paradise. They're looking at the problems in a fundamentally different way than you or I are.