I agree with your advisor. With the "bottom up" approach you can study mathematics all your life, and have great fun in the process, but to start a successful research career you need to focus on a specific problem even if you do not know everything about it yet. In fact, many people learn much better that way, motivated by specific problems, which is why the earlier one starts doing research, the better.
Becoming a well-rounded mathematician will pay off in the end but at the moment you just need to stay in the game. There are indeed some remarkable people who can cover different areas with continuous research output, but for many of us this strategy won't help to find a job, get a first grant, and honestly, won't help to make a significant contribution. There is good reason mathematicians specialize: one deep paper has more impact than 10 mediocre ones. You have to really get noticed in some subfield in order to succeed careerwise.
Again, there are a number of prolific people doing deep work, and if you realize you are one of them, stop worrying. Till then, it is better to focus.