Agile Common Understanding
In short, agile methodologies like Scrum or XP usually split responsibilities into 3 parts:
the cross-functional team : all the people (developers, testers, integrators...) needed to achieve actual project work. They usually are self-organized, which basically means that there is nobody outside the team who assigns work to individual team members. They commit themselves on the work they can do in each iteration (for iteration-centric methodologies like Scrum. It might be a bit different with flow-focused teams like in Kanban)
the coach : called "Scrum Master" in Scrum, he is responsible for the HOW part. He knows the methodology and how to apply it. His work consists in helping both the product owner and the team work together. He is not a project manager in the classical understanding of the term and will not assign work nor impose his decisions.
the product owner : called "customer" in XP, he is responsible for the WHAT part. The product owner is the stakeholders' representative. For the team, his role consists in providing a prioritized set of items to work on (features, user stories, etc.). For the stakeholders, his role is to report project progress. The product owner usually works full-time on the project and is easily available for the team.
So if you - I mean your company, team, PO, etc. - explicitly decided to use such a methodology - but you did not - there should be no doubt about what is everyone's responsibility. In this case the Product Owner should not (have to) deal with the HOW part at all.
The role of a Scrum Master is important here as he will make sure that nobody crosses the line. For a Scrum-focused answer you should also read the question "Why can't the scrum master and the project manager be the same person?" which features very interesting answers.
Role definition
my 20-person development team gained a Product Owner
My understanding of your question is that the Product Owner was kind of dropped into your team deus ex machina. If I understand well then there might be a misunderstanding in role definition that you must try to clarify. David Espina's answer is pretty clear here on this point. "Product Owner" is just a name after all...
Classical Project Manager
The PO wants to know how many hours programmer X has available during each sprint and exactly what he will do with them.
If your PO used to be a command-and-control project manager, he might have trouble with becoming "only" a PO. He is used to having lots of metrics related to individual work that allow him to monitor the progress of the project. Try to understand why he wants to collect these data and how you could provide him relevant indicators that would achieve the same role but at a higher level (velocity, throughput, cycle-time, etc.)
He might also want to make sure that everybody is 100%-loaded. Please refer to CodeGnome's answer on that point. Also read Pawel Brodzinski's article A Myth of 100% Utilization.
Trust
"OK, we have 40 person hours of work for Arthur and 32 person hours of work for Candace, and since they're each supposed to be half-time on this project, they should finish that work by the end of the next sprint."
If your Product Owner often wants to go to this level, then there might be a trust problem between him and you. Maybe your lack of experience with Agile projects, his experience with non-Agile projects or a possible broken commitment earlier in the project makes him reluctant to trust you and your team. Under business pressure, he might want to "do it himself".
In this situation you want to sort things out quickly, since this lack of trust is likely to kill your project.
Summary
Is my statement above about the PO's role legitimate?
In absolute terms : yes. Your definition of a Product Owner is very "scrumy". But the thing is that there obviously is a misunderstanding somewhere, either on each other's role, or on each others skills, or on each other's goal, etc. Make sure that you both are on the same page on this question.
