I have PayPal, Amazon, LinkedIn, <<mid size company>>, <<mid size company2>>, <<startup>> as a manager on my resume. I didn't get a call back after applying for two weeks.

I got 4 or 5 standard rejections.

I have non-English name so that definitely hurts. I have AP EAD which is a stage between H1B and Green Card and I still require sponsorship. It's complicated but I can't just switch to EAD right away.

It's not just engineers. It's managers and experienced people as well. Don't believe top comment that it is bimodal. Unless you are supertar (99.99%) it becoming hard to get noticed. I thought of going back to IC role but it is hard to pick up and do leetcode all over again. It is extremely hard with a special needs kids at home.

Any suggestions or recommendations for me?

I understand this a lot.

I had managerial* position I ghosted because they had Leetcode literally written on the agenda.

* - managerial is replaced with Lead. Lead is expected to be hands-on as well as have serious managerial experience. Since it's easier to lie about managerial experience, you have people lying into these roles and becoming terrible managers.