Why should I enter a pure math PhD program if I most likely won't end up in academia?
why should I enter a pure math PhD program with the goal of entering academia if I will (1) probably not achieve this goal, (2) probably have a few hard years after my math PhD, and (3) probably end up in industry in which case I could have taken a less stressful route and even prepared myself better with a PhD in another field?

With the mindset that you are showing by the very loaded way your question is phrased - including several rather misguided beliefs I will address below - consider the possibility that the answer is “you shouldn’t”.
What are these beliefs I’m referring to? They are:

Something is not worth trying if the probability of success is lower than 50%

The definition of “success” in pursuing a pure math PhD is to have a career in academia

Any argument for pursuing a pure math PhD is not helpful or worth considering if it is “vague and subjective”.


In fact, these beliefs are not true in any universal sense. The decision to pursue a PhD in pure math is a personal and completely subjective one. It’s right for some people, and not right for others, and for each person it’s either right or not right for different reasons: e.g., for some people who do define success as an academic career it’s still the right decision because they are talented enough to defy the average not-so-great statistical odds you are referring to; others simply define success in completely different ways.
Likewise, the people for whom the decision does not seem like the right one also differ in the reasons why they reach that conclusion. Some are passionate about math but don’t think the effort, stress and opportunity cost are worth the risk of not achieving the dream of an academic career. Others may be so mathematically talented that they can know with a fairly high degree of certainty that they can have an academic career, and yet nonetheless find other opportunities more alluring.
Your question suggests that you want someone to come here and persuade you to do a pure math PhD by giving you some objective, scientific reason to do it. That’s not how it works. You’ll have to do your own thinking and make your own decision based on your own ideas of what matters most in life and what kind of life and career you want. I said at the beginning of the answer that it sounds like a math PhD may not be right for you, but of course I don’t know you and may well be wrong about that impression, so don’t let it prevent you from making the choice that’s right for you. Good luck!