It doesn't beg the question, that's a logical fallacy referring to circular logic - it raises the question.
Defining utility is an axiom in itself, and that's where we can have a discussion. In modern society, it can be argued that all high utility subjects share the same common theme - they lead to social positions that are essential for the functioning of society. An example might be applied physicists, in other words engineers, because every part of a first world citizen's life revolves around them performing their jobs correctly.
The dwelling I presume you reside in, needed people to calculate whether the structure will fall (especially taller buildings). The potable water that comes into your home, requires people highly skilled to decontaminate and filter and process and pressurize and deliver the water. The food you eat, as we're no longer an agrarian society, require modern machinery and industrial scale processing to sustain.
This could go on ad nauseum, but the point is STEM offers students a chance to develop the analytical skills required to go into these highly essential, sought after jobs. If economies fail, and anarchy begins, then these people will no longer be in a position to do their jobs properly and we both know what happens to social order then.
These are all examples of high utility.
The less needed, but equally valued, would be artistic works. The wealth of ancient empires lead people to paint, and sing, and enjoy life - that's great, but not essential.
The comments about vocational courses - I consider them a subset of STEM, through an develop-experience-with-on-the-job-training route rather than a highly analytical route. It does depend on the nature of the vocational course, some of those jobs (if menial) are only available because we haven't figured out a way to make a machine do it faster and more cheaply.
Personally, I can think of many graduate level jobs that (in effect) require STEM degrees. It's pretty difficult to work in Chemical Engineering without some graduate level background in chemistry, or Doctor without some graduate level background in medicine, or NASA scientist without some graduate level background in physics/engineering. A reputable lawyer without a law degree.
This could go on ad nauseum.
There's no one denying that experiences/transferable skills are perhaps more essential than the degree in itself, but the graduate level background is the starting point for the higher end knowledge labourer jobs.
The happiness topic is far beyond the realm of TSR, but I will comment this: economists/psychologists do study correlations between overall happiness in life between people who have high income and those that don't.
It increases until about £60,000.00/year (ie all the material goods necessary for a comfortable life with family), then plateaus and falls (the overall stresses and responsibilities of very high income persons, with exception of passive income)