Outages during scale out, weird insurmountable restrictions on cross region replication, cost, backup and restore doesn’t work properly, cluster irrecoverable corruption and it is 100% opaque whenever something goes wrong. This results in escalation to AWS support who barely understand it.
The big sell was easy deployment and cost management but the negatives outweighed that.
We tried Elastic themselves but the sales and technical process was a nightmare. It was virtually impossible to work out what we were going to have to spend. We needed X-Pack (RBAC and SSO) but the enterprise licensing is all over the place for it.
Ergo we didn’t bother and stuck with our DBMS FTE stuff. It’s crap but the ROI is better.
Part of the issue is lack of visibility into what is going on, you don't get a lot of ability to debug.
Company I was at switched from AWS's ElasticSearch to hosting it ourselves on top of EC2 instances so that we could control the memory/CPU/disks and see logging/understand more about the internals to help understand why it was slow for our use case.
Edit: See my comment further down for an extrapolation. This one lacks merit which was a bad call on my part.