In case anyone else still encounters this issue even when applying the correct RBAC permissions, we had a difficult to diagnose edge case that led to the same symptoms.

We have a use for both System and User Managed Identities assigned to the Logic App (Standard). Our Bicep templates were assigning the above RBAC permissions correctly to the System Managed Identity principal. We also had AZURE_CLIENT_ID set in the Logic App environment variables with the identity of the User Managed Identity so that this could be used as the principal for auth with certain other services.

It seems that the Azure Blob Storage SDK being used by the Logic App internal connector picks up AZURE_CLIENT_ID if present and uses that identity for its authentication with the Storage Account (which in our case, didn't have the "Storage Blob Data Contributor" RBAC permission set because that identity was not intended to be used for that purpose).