Yes, you absolutely want your PO at the Daily Scrum.
First off remember that the Scrum Guide is something like 17 pages and only covers the broadest brush strokes.
The product owner is part of the team. They are the direct connection to the business and the person who signs off on stories as done. You absolutely want the PO there so they know what's going on.
The intent of the Scrum Guide is generally agreed to be that only the developers are talking and the PO would hold questions until after the standup. Most agile teachers recommend that the PO only ask brief questions during the standup, to clarify things. If a story is done, the PO is able to follow up with the team member after the Daily Scrum to approve the story, so it can be properly moved to done.
Interestingly enough, Mike Cohn just issued some advice on this in his direct email letters to subscribers. He recommends that everyone, Scrum Master and Product Owner included give updates in the Daily Scrum. It's an evolution of our thinking which recognizes that there is other work, beyond the coding, that happens during a sprint.
And remember, it's all guidance. If it works for you, that's the most important measure of if you should do it.
EDIT: I'm in the process of gaining the Certified Scrum Trainers certification. So I'm explaining this from that perspective. The Scrum Guide is a framework. It is not prescriptive, it is a guideline. It is also an ever evolving concept. The last time it was updated was more than three years ago and Jeff Sutherland, one of the authors, has certainly continued to evolve his opinions.
So if anyone tells you "You're not following Scrum by the book" then they are the ones with the problem. A 17 page document cannot and was not intended to ever be the be all and end all.
Product Owners absolutely should be at the Daily Scrum. Every CST I know teaches that. The exact role of the PO at the Daily Scrum is up to the team to decide based on what they feel works. Remember, inspect and adapt. We don't hold rigidly, we do what works.
So if i was your coach, I'd tell you to start with the "by the book" only the developers talk. Then use your retrospectives to decide if that works.
A couple of other notes:
- Jeff Sutherland and Ken Schwaber maintain the Scrum Guide. They are only two voice in the scrum community and not the only ones who were there when it was invented.
- The Agile Manifesto isn't Scrum and it is the over arching guidelines for us all. Inspect and Adapt is a key principle. As is individuals and interactions over process and tool.