Not every student would benefit from learning set theory early beyond the universally needed understanding of injective/bijective mappings, but some would. It does depend on personality. It has some relation to cultural things.

"Role of mathematics" implies relatively long run; experience is felt in a very short run.

If you want to extend your intuition into an area nobody understands well, you often need to combine quite weak analogies and formal methods - because you need to do something to get any useful intuition.

There are many branches of science where you can get an amplified feeling of understanding somthing in the area where you don't have working intuition. There are three culture-related questions, though. First, how much (true or fake) understanding of facts you get from the culture before you know the truth? Second, how much do you need to learn before you can understand a result surprising to you? Third, is it customary to show the easiest-to-understand surprising result early in the course?

Of course, for different people in different cultural environments different areas of maths or natural sciences will be best. But it does seem that for some people the easiest way to get an example of reasoning in intuitively incomprehensible (yet) area is to learn set theory from easily accessible sources.