Security wise, as far as I know AWS isn't offering anything additional here.
The argument of "but you're paying to not have to worry about it!" that is a fair argument. But comments like this seem to purposely over-state the difficulty of running a not-super-complicated tool like Grafana in order to make "only $600/month, or $7.2k/year", sound more appealing.
Moving to a different server (or cloning for HA redundancy) takes maybe 5 minutes to copy the JSON and make a crontab entry. It's pretty much the lowest headache software imaginable.
They follow the 12-factor app principles, provide a ready-to-use docker image, support various auth providers ( my company uses GitHub), integrates well with AWS IAM, etc.
As someone else said here, you just have to use RDS to store the state.