Both AWS and Azure's managed SQL Server database solutions are hobbled like that. They can be replication targets but not sources. At least RDS for SQL Server

Both AWS and Azure's managed SQL Server database solutions are hobbled like that. They can be replication targets but not sources. At least RDS for SQL Server lets you do native backup/restore, so while it's slow there's still a good way to get your data out. Azure SQL, you're stuck with something like the Elasticache solution. (Or theoretically BACPACs, but they aren't transactionally consistent and don't work for large databases.)