Only allow the creator of the item and the user direct manager to view the submitted item

I think the new List Form will get what you're after.

You can have the list setup so only admins have access to it. Then you can create a List Form that allows users to submit items to the list - note that when using the new List Form the user doesn't need access to the list to submit items. The user just needs the link to the List Form, so you could add this as a Quick link on a SharePoint page.

Then, once the item has been added (only admins have access at this point) you can run your Flow to add permissions (read) to the user that submitted the item, and permissions (approve - or similar) to the user's leader. Then, if the leader approves it, reset their permissions to read, and add the next leader with the appropriate permissions, etc.

This should provide everything you're after based on the requirements you listed in this post.

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