You can't just rsync files into a fully managed RDS PostgreSQL or Elasticsearch instance. You'll probably need to do a dump and restore, especially if the source machine has bad disks and/or has been running a different version. This will take much longer than simply copying the files.
Of course you could install the database of your choice in an EC2 box and rsync all you want, but that kinda defeats the purpose of using AWS and containerizing in the first place.
That is true, albeit not in all cases!
An alternative approach (that has some serious caveats) would be to do full backups of the DB directory, e.g. /var/lib/postgresql/data or /var/lib/mysql (as long as you can prevent invalid state data there) and then just starting up a container/instance with this directory mounted. Of course, that probably isn't possible with most if not all managed DB solutions out there.