You could equally say that everyone who looks at the rules of chess sees the same thing. In order to show some inevitability to ZFC, you have to show that unconnected parties arriving at it independently.
On the one hand, why? I'm quite happy to say that chess exists. Not everyone will ever see chess, but not everyone will ever see Neptune. Among all the games that could be played, chess is but one grain of sand on the beach. But the grain of sand exists regardless of whether anyone sees it.
On the other hand, there has been, I believe, a substantial tendency for people devising alternative axioms for the concepts of sets to come up with things equiconsistent to ZFC or to subsets of ZFC, and with fairly direct translations between them. Compare also the concept of computability, where there is a very strong tendency for different ways to answer the question "what is computation?" to come up with equivalent definitions.
It is (I think) true that if you try to come up with an alternative foundation for mathematics you are likely to get something that's equivalent to some subset of ZFC perhaps augmented with some kind of large cardinal axiom. But that doesn't mean that ZFC is inevitable, it means that if you construct two theories both intended to "support" all of mathematics without too much extravagance, you can often more or less implement one inside the other.
But that doesn't mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + In...
Among my friends interested in rationality, effective altruism, and existential risk reduction, I often hear: "If you want to have a real positive impact on the world, grad school is a waste of time. It's better to use deliberate practice to learn whatever you need instead of working within the confines of an institution."
While I'd agree that grad school will not make you do good for the world, if you're a self-driven person who can spend time in a PhD program deliberately acquiring skills and connections for making a positive difference, I think you can make grad school a highly productive path, perhaps more so than many alternatives. In this post, I want to share some advice that I've been repeating a lot lately for how to do this:
That's all I have for now. The main sentiment behind most of this, I think, is that you have to be deliberate to get the most out of a PhD program, rather than passively expecting it to make you into anything in particular. Grad school still isn't for everyone, and far from it. But if you were seriously considering it at some point, and "do something more useful" felt like a compelling reason not to go, be sure to first consider the most useful version of grad that you could reliably make for yourself... and then decide whether or not to do it.
Please email me (lastname@thisdomain.com) if you have more ideas for getting the most out of grad school!