Oracle DB is, or was, very close to that with its query profiles, baselines, and query patches. It wasn't automatic back in 2014 when I last worked on it, but all the tools were there. Heck, it was possible to completely rewrite a bad query on the fly and execute a re-written variant. I suppose it all stems from the fact that Oracle is regularly used under massive black boxes, including the EBS.
Also, the problem with automatic indexing is that it only gets you so far, and any index can, in theory, mess up another query that is perfectly fine. Optimizers aren't omniscient. In addition, there are other knobs in the database, which affect performance. I suppose, a wider approach than just looking at indexes would be more successful. Like Ottertune, for example.