LeadSync accesses your leads via your personal account, not your ads account.

So long as you are an admin of the Facebook Page you're using to advertise, we will be able to send lead notifications when a new lead is created in an ad using that Page.

This is just the way Facebook set lead access up when they first created the product. By default, individuals who are Page admins have access to leads created via that page, NOT the Ad Account.

There is a way to set up access via Business Manager, but it is a little more complicated.
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