I feel like I have just lived through this on the supervisor's side.
I had a student who did not complete tasks and was generally very difficult to supervise. Eventually, I had to tell them that their behavior was unacceptable.
I found out that:
- They did not get into the program they wanted, and I was their backup solution.
- They consider my research second-rate.
- They intentionally ignored my advice because they consider me a bad researcher and believe it is in their best interest not to follow my footsteps.
This student felt justified in their position because, during their time under my supervision, they were not able to publish. They were resentful towards me, believing it was my fault they had not published.
In retrospect, I understood that it was a vicious negative feedback cycle. I was unable to help them publish because they would not heed my advice, which led to them listening to me even less and provided me with even fewer opportunities to support them.
(For the record, all my other students publish in the top venues of our field of research).
Needless to say, this student is no longer under my supervision. It is impossible to supervise someone who does not respect you. The whole experience was painful and a waste of time for both of us.
Do yourself a favor and either:
a) Politely decline.
or
b) Acknowledge that you are in a position where you have a very limited view of the field, and that other people, with a broader view of the field, trust this person enough to provide them with funding to hire you. Find out what this professor does well, and learn to appreciate their work. Find out how they can support you. Once you have identified the professor's strengths, reflect on their weaknesses again, and figure out if they can still be valuable to you. Maybe even in conversation with your professor. Don't say "you are bad at X," but say "I noticed you do not focus on X, but X is something I struggle with. Would you be open to seeking collaboration with an expert in X, as I wish my PhD to include X? "
The relationship with your supervisor is essential especially during the early phases of your PhD.