tl;dr I was ignored for not having a US visa, even though I said I'd need visa sponsorship multiple times. Total waste of time.
When submitting your application at Amazon you have to check a box signaling whether you have permission to work in the US or I'd you need visa sponsorship. I checked the box and applied.
Two weeks after I received an email from Amazon about the first step, which for me was directly the technical phone screen. As asked in the email, I sent my time availability, timezone, and explicitly said I'd need a visa to work in the US.
Then I moved to the on-site stage, with another automatic email from the recruiter asking the same things. Again, I told them I'd need visa sponsorship. In the middle of one of the technical interviews, I was asked by the interviewer if I had permission to work in the US, and he seemed quite surprised that I didn't. He asked twice if the recruiter knew about this and I answered.
I got nervous during some of the technical interviews and didn't do well but I aced the behavioral ones. A week later the recruiter called me and told that even though I wasn't accepted for the Senior Data Engineer role, they would like to offer me a Senior BI Engineer role. The recruiter would look for open positions and get back to me, and I would just need to talk with a manager.
As I knew the yearly H1-B visa deadline was approaching, I asked about how that would work, and she seemed surprise that I didn't have a visa already. I told her it was in the application and the emails, and she told me she would need to talk with the visa area to see about the arrangements.
Two weeks later and I didn't receive anything from the recruiter, so I sent an email asking about any updates. She sent over in writing everything we talked about how I was rejected for DE but accepted for BIE. She also asked me if I could lookup roles that I was interested in directly from the website, and I did that. Never heard back from her again.