How to get more time with the manager?

I understand your concern. Here are a few things to consider:

1. Does every direct report get the same amount of 1:1 time with the manager right now? If yes, it doesn't solve your problem but at least tells you that you're being treated equally among all their reports. :)
2. How do you want to use the additional time with them? I'd get clarity on this first. Identify your needs, how quality time to discuss those topics would actually help you, and then have a conversation with your manager focusing how the help would empower you.
3. Lead the conversation with "I'm looking to get X from our time together. X will help me in Y way. How do you recommend we do this?" The manager might ask your more questions at this point, offer more 1:1 time, or suggest other options. I'd say be open to other options as well. At the end of the day, you are looking for Y. It doesn't sound like getting more time itself is your goal. Asking an open-ended question will probably be more productive than asking for more time upfront. You also don't want the manager to get defensive and say no (which will close the conversation preemptively!).
4. Can you make better use of the time you do have? Like have a shared document where you add your top of mind the day before and have an agreement with the manager to respond on the doc. This will give you the full 25 mins to have the conversations important to you.

Hope this helps.

Shine hit the nail on the head with, "How do you want to use the additional time with them?"

If you have a concrete discussion topic and do the legwork for it, then I'd request a dedicated conversation with your manager. This is pretty common with performance review discussions.

After you get their buy-in, you can send a separate calendar invite where the title makes clear the agenda topics. I talk about how to uplevel your 1:1s here: https://www.jointaro.com/course/the-complete-onboarding-guide-for-software-engineers-succeeding-when-youre-new/uplevel-your-11s/

25 minutes means that someone is showing up late (I assume the manager). I would give them constructive feedback about that. 1-2 minutes is okay, but 5 is too much (that's 16.6% of the time!).

10 minutes for top-of-mind things seems too long as well, can you cut it down to 5?

Lastly, Shine's 4th point is the core tactical advice I have here (it's super easy and has a high chance of meaningfully fixing the problem). Seed the agenda with meeting topics beforehand and let them know. We have a short video about that: A Simple Trick To Make Your Meetings More Efficient

As an E6, you are incredibly important to the team. If you put in the effort to make the meetings more proactive/streamlined while delivering the feedback, you should be able to greatly improve the situation. 11 reports is pretty ordinary for an EM, so that's not a problem either.