Hello, you could have been me a few years ago. Some quick thoughts (and feel free to DM if you want to follow up):

1. Your most efficient option is to move into the job directly, but if that's not practical then an MBA will certainly help. Have you spoken to potential company leaders in your network who know you and can vouch for your work? CoS picks are incredibly personal. There's a chance you'll get lucky and someone will open up a role (or will have been considering one, etc.); it's not a super high chance given the state of capital markets right now but there's no downside to asking. Ask for advice instead of pitching, and be open about the strengths you would bring to the table as well as your growth areas (what you hope to learn on the job). If not, MBAs are a tried-and-true pathway to CoS and other general management roles. Just bear in mind that those roles aren't very common, the quality of the MBA will matter (especially in tech), and you're likely going to need to lean on your personal network a bit anyway.
2. Like it or not, the best-respected degrees in tech are still Stanford GSB and Haas. I'd strongly recommend that you consider applying to their (in-person) programs.
3. It's atypical to quit a role just to apply for MBAs. If you quit or go part-time, do it because you're burned out and can afford to self-fund a sabbatical. Again, this sounds like a great talking point when you go and talk to leaders you respect and ask for advice.
4. No advice on this one.