Your manager's got this. It's not on you to figure out how to fix a skill gap, unless your manager gives you that (manager's) assignment.
In your conversation with your manager, let them take the lead and ask questions. Then ask your manager to keep confidentiality. Then answer the questions truthfully.
Depersonalize the issue. It's about ...
How can I phrase this in a way that's professional AND doesn't hurt Joe's reputation too much AND doesn't make me sound conceited?
I would suggest not to get into the meeting discussing your Root Cause Analysis of the problem (it seems that Joe really just doesn't have the skills at this point to lead the project).
There can be any number of reasons for ...