"How much would you subscribe to the statement that studying questions one finds interesting is something established mathematicians do, while younger ones are better off studying questions that the rest of the community finds interesting?"

Not at all.  I don't think anyone, young or old, will find success by working on questions other than those they find interesting.  Mathematics is just too difficult for that.

Ideally, everyone should work on problems that are interesting to both themselves and the community.  Senior mathematicians have the luxury of working on problems whose interest to the commmunity has not been established.