AAS Can't Connect to Azure SQL Managed Instance Public Endpoint

Question

The documentation says AAS supports ASMI public endpoints as data sources, see https://learn.microsoft.com/en-us/azure/analysis-services/analysis-services-datasource

We have enabled a public endpoint on our Azure Sql Managed Instance and allowed traffic from "Azure Cloud Service", all in accordance with the instructions here: https://learn.microsoft.com/en-us/azure/azure-sql/managed-instance/public-endpoint-configure#allow-public-endpoint-traffic-on-the-network-security-group

We connect SSMS to our AAS instance, open the Connection properties dialog for our AAS database, open the connection string properties, enter the ASMI public endpoint as the server name, and enter the name and password of an ASMI login. If we hit the Test Connection button, the test succeeds. The AlwaysUseGateway property is set to False, and Impersonation is set to Service Account.

Then we close the Connection properties dialog and try to Process the AAS database. We get this error almost immediately:

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Failed to save modifications to the server. Error returned: 'On-Premise Gateway is required to access the data source. Please install a unified gateway for the server XXXXX or disable AlwaysUseGateway setting if it's enabled.

Technical Details:
RootActivityId: abfb86f5-1405-4d69-a071-fb9b3e613646
Date (UTC): 8/10/2021 4:06:24 PM

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Why are we getting this error?