Being a mathematician or thinking like a mathematician is about being confused and trying to cope with it. This is the typical situation.
Usually, you have some idea of what you should be doing or know and you can work along that line incrementally.
Thus learning from established sources, applications, or developing new math proceed along similar lines.
Existing materials often have substantial gaps. These become larger as the courses or subjects become more advanced.
A major reason for this is lack of a critical math. There is insufficient supply and demand for filling the holes.
Advanced university courses are typically full of holes in their materials. They often do not have enough simple examples. They don’t build up more complicated examples. Solved problems are often available in a cryptic form that is close to useless for outsiders and often most insiders.
The amount of pretense to peers, students and users is substantial. These gaps are passed off as not existing, trivial, or the fault of the user in perceiving gaps.
This is not unique to math or to universities. It is true in government and industry. There is much work to be done in any of these spheres.