How can I convey my concerns about a University graduate program to the director without sounding like a know-it-all?
Kudos to you for trying to improve the lecture! I was in comparable situations myself, but I'm a coward and didn't take steps. So this answer deals with how I would go about it, if I had to - I didn't test it in practice.


  However, these questions lead to more questions, comments, chatting, arguing, even talking about politics with the lecturers, to the point that the lecturers can't finish on time.


This is the important point and what you should focus on - the clearly detrimental effects (discussions/arguing, not being able to finish on time).

For sure, the usual etiquette applies


Be polite and friendly
Be to the point
Be concise
Keep it well-arranged


This applies to both, talking and email, e. g. in the latter case I try to write them in such a way, that what I want becomes clear already by glancing at the email. No unnecessary fluff.

I guess I would take the following steps


Praise the lecture (if you mean it, but you'll surely find something positive to point out) and especially the possibility to ask questions.
Describe how the way questions are handled interrupts the lecture, but not in an accusatory way. Say that it negatively effects you also.
Add suggestions on how to improve the lecture, e. g. by answering the questions directly only (and to cut discussions short) and your suggestions.


Explanation:


Gives them a good start and also makes clear, that you are neither just whining/complaining nor question the usefulness of asking/answering questions itself. After all, this is one of the advantages of attending lectures.
Explaining the problem situation and the effects it has on you, without blaming them directly, gives them valuable feedback. It's possible that they are already on your side, but if no one complains... Also, it is hard to not agree with you and the effects it has on you cannot be denied. How and that the learning experience suffers should be of great importance to them.
Suggestions are always helpful, and in this case you further emphasize, that you have no problem with questions themselves and are therefore not advocating something bad or having unrealistic expectations.


For sure, it strongly depends on them, how they react to your suggestions. It's possible that they don't do anything at all, or they may welcome them as an incentive to act. In the end, you are there to learn, and interrupting the lectures is detrimental to the whole purpose of such lectures, so it's good that you voice your opinion on that matter. Just avoid making it sound as if you were accusing them personally, rather that there is a problem and you needed their help to solve it.

I have good experience with constructive and polite feedback and my lecturers always seemed willing to listen to suggestions (usually by others, not by me) and actively encouraged feedback. The feedback was not always helpful (e. g. I noticed that, when the (anonymous) comments were discussed), but whenever it was, the lecturers were open to it, or at least explained why it wouldn't work.