What are the benefits and drawbacks of fitting one's problem within an advisor's research?
With regard to choosing to do your PhD research with a group working in your field of interest:-

Benefits:

In-depth subject specific knowledge, meaning that:

You will get started faster.

Your early mistakes and false leads identified and corrected earlier.

You will get a faster entree into the best collaborative networks.

You will get noticed by the community faster.

You will be associated with experts in the field.

Disadvantages:

Depending on the supervisor, you might get rail-roaded into doing the science his/her "way", leaving less opportunity for you to strike out on your own path.

You are dealing with a new advisor, and that relationship takes time to cultivate.

For your situation, it seems that you will have to get a source of funding.