<But what would happen if we find a way of removing an incompetent president? That won't solve the whole problem.>
The Constitution offers several avenues for removing a president.  It's not easy to do, nor should it be; and it absolutely requires courage and integrity among the leaders and office holders of the president's own political party.  We saw this during the Watergate crisis; we have not yet seen it during this presidency.  I think we will.