How can I ask a teacher to follow the textbook more closely?

You may certainly ask -- politely! -- but you should expect to be told "no", and you should accept that answer if it is given. In most graduate courses, it is the privilege of the instructor to decide what are the important points and how to organize them.

If the course is preparation for a qualifying exam (for instance), then there may be a more rigid syllabus that needs to be adhered to. In such a case, if the instructor were deviating strongly from the expected objectives it would make sense to complain.

As an aside: since the book presents things so well (from your perspective), you can learn the material from it and consider the lectures as supplementary. It then seems to me that having lectures that approach things from a different perspective than that of the text is a benefit. An instructor who parrots the text would be redundant.

Wow. I deliberately "leave a lot of gaps that the book covers nicely" in advanced courses. I say "this is in the book." And I actually aim to leave them especially in the parts that the book does best.

Now, I always ask for any questions from the class, about anything I say, whether it is in the book or not. But I do believe advanced students should get what is in the book from the book unless they have specific questions they want to raise in class.

So, if your teacher does not even take questions about textbook material that he does expect you to learn, then that is your problem. Ask the teacher to take questions -- and of course ask it politely.

If the teacher does take questions, though, then rather than ask the teacher to better anticipate your questions maybe you should work on asking more questions yourself.

I'm seeing a general disdain for postdocs (who are simply more junior than the professors; one day, some of them will be the hotshots of your field), and I can't decide if your feeling superior comes from the actual bad teaching, or from your disdain. Are you sure that the postdoc is omitting important results? Are they important in your opinion, or did you get this information elsewhere? In any case, a bit of respect would go a long way, when you ask the "postdoc" to follow a single reference.

I am more concerned by your attitude than your professor's plans, however. It is quite common to not recite an entire book, and pick and choose the important topics from a text. This leaves room for your proactivity: if you want to learn the material in more depth, you can do so by reading the text, including the part the instructor has left out. That's really what grad school is for. You take control of your own studying.

That being said, most postdocs are very receptive to feedback, as they have not been teaching for very long, so if your comment is a valid one, and if you can do it without hurting his feelings, he should be open to your suggestions (however, don't say things like "you should sacrifice the material you intend to cover for clarity", as the power to decide on the syllabus rests entirely with the postdoc.)