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It looks like the pattern you're describing is most likely a Kerberos double-hop issue rather than a permissions one.
When the package runs on the SSIS Catalog, your credentials reach the SQL Server but aren't forwarded on to the file share. So the connection arrives there as the SSIS service account, not as you even though the SSISDB log shows your ID as the Executed As Name. That's why your account has access but it still fails.
To confirm, you can have your Script Task log that shows the actual identity hitting the share.
A quick test is to grant the SSIS service account read/write on the share. If that fixes it, the longer-term options are running the package through a SQL Agent Proxy account that has share access, or configuring constrained Kerberos delegation with help from your AD team.