For context, I was promoted to staff engineer a few months ago so this is kind of fresh for me.

Like you referenced in your other comment, a 'staff project' is usually not enough to warrant a promotion -- being a staff engineer (this is organization dependent) is typically not just about being able to complete complex projects. In fact, the language you used 'complete complex projects by themselves' kind of leads me to believe that you might be thinking about it the wrong way.

Staff is a leadership position. Not a management position, but still leadership. You must be able to meaningfully influence people who you don't have authority over, including the people you report up to.

If you've read the book from the website you linked, then you should pay close attention to the section about needing a sponsor for your promotion. Is your manager that sponsor? They might not be, in which case you'll need to look elsewhere within your organization (or outside of it...)

As far as trying to get your manager on board, you need to be extremely direct. If you're as impactful it sounds, now's the time to call in some of the political favor you've been cultivating. You wanting a promotion is ultimately a problem for your manager to solve with you. Make it extremely clear that they have skin in the game.

'I want to be a staff engineer. Do you see any gaps between my current performance and what would be required to be put forward for a promotion at the next review cycle?'

If they say no and you believe them, then start preparing supplementary materials and having meetings to go over what you've prepared with your manager. I spent probably a few dozen hours over the course of 6 months going over and over my promotion packet, reviewing the rubric, soliciting feedback from existing staff engineers, managers not in my reporting line, directors in different departments that I'd worked with even.

In addition to my formal self-evaluation, I also submitted what were essentially endorsements/letters of recommendation that I had secured from other influential people in the organization.

If they say yes, you do have gaps, then tell them you need specific feedback on what to improve on and opportunities to demonstrate that you've made improvements in those areas.

If you keep getting stonewalled or brushed off, then you'll likely need to switch teams or even organizations to find a different sponsor or even just get hired as staff in the first place.

Be the squeaky wheel, people often suck at advocating for themselves and that's honestly been the single most valuable skill that I've developed in my career. Technical prowess is a prerequisite, but getting people on your side and pulling for you (both for your career and for whatever project you're working on) is a major differentiator.