I'm not going to gainsay your experience, but it doesn't match mine. Much of what you do in graduate school is to discover where the accessible areas of research are--where do we have a foothold, and are making progress, and what are some achievable results?

There are big and hairy problems that are bad investments for a young mathematician. I would steer students clear of the Collatz conjecture. But once you get up to speed in your research area, you usually find interesting problems thick on the ground.

Tenure-track positions are competitive, but I don't think there are a lack of interesting things to work on.