Ask HN: What Are Good Product Manager Qualities
How important are design skills for a product person? Is it more important to have engineering background or good design skills?
What you didn't mention, and what I think is very important in a PM, is "people skills". PMs have to be able to interact effectively with the sales, marketing, and support organizations, and also with users. They have to be able to envision themselves in the position of the user, and have empathy for users problems.
(I'm an engineering manager who has worked closely with many PMs over the years.)