We're talking much more than just index optimizations. Think automatic OLTP vs. OLAP pattern detection and migration of data into columnstores or in-memory tables. Automatic detection of cold vs. hot data and migration to-from expensive fast near storage (SSD) vs. cheap remote storage (S3 like). Automatic resizing of instance size. Materialized views. On-the-fly query rewrite. The field is wide open. And you have the best experts in the field looking at your database.
By contrast, the traditional on-premise auto-tuning did not made much progress over the past 10-15 years. There are reasons for this: small operational dataset (one site only), high risk of regression (difficult to do A/B testing), scarcity of experts. The best avenue for on-premise I predict will be the trickle-down effect of migrating service know-how and technologies to on-premise offerings. We're also going to see a push up the stack of the know-how and experience from operating the service making its way into the app stack (think EntityFramework v. 2018?) and helping developers avoid the common pitfalls.
BTW, only 3 days ago Adrian Coyler's morning paper discussion was Self-driving database management systems [2]
[1] https://docs.microsoft.com/en-us/azure/sql-database/sql-data... [2] https://blog.acolyer.org/2017/01/17/self-driving-database-ma...