Intuition is a part of the mind that makes approximate models based on experience up to that point. The experiences without anything that would help model infinites would make them non-intuitive. Experiences that would help model infinites would make them at least more intuitive. Experience with rational investigation of them going through examples is actually one way to do that. There's been organizations like Marine Corps incorporating intuition into training for a long time where they set out to program it using realistic scenarios in drills that people do over and over until the intuitive processes pick up the patterns/model. It will happen as many people study that topic as well.

The trick is that you need good data to feed it that makes the patterns clear. You might have not gotten a lot of that on the topic you had no intuition of. On some topics, there hasn't been much of an attempt to do that since people expect it to be hard with everyone forced to do the methods that are non-intuitive. Lots of traditions are like that. Then, there's the possibility something can't be intuitively handled at all. I don't know if those exist but they might. That many activities of people doing math seem to get an intuitive boost during their day-to-day work makes me lean towards intuitive possibilities for about everything that can be approximated (esp with heuristics).

A chess master can intuitively tell when mate is three moves away. They weren’t born with that intuition, though. They gained it over time.

You aren't born with all of your intuitions, you learn most of them. This includes mathematical intuitions.

At this point a mathematician might step in and say that you all need to define your terms. You're basing your arguments on two subtly different definitions of the word "intuition."