You can get it down to $12/month by going with a t4g.micro (2 vCPU, 1GB RAM), single-az, no proxy, no backups, 5GB storage; with price after that scaling roughly linearly with RAM. But the interface will scream at you the entire time that this isn't reasonable for a production setup.
It is really no different than managing VPS's, in all honesty (just less expensive and tangible). I spend less than one hour a week doing updates and routine maintenance. Drive failures happen rarely, and can usually be swapped out within a few hours (this is what RAID is for).
More time is spent dealing with k8s and it's shenanigans than dealing with the actual servers.
Many people don't do that. My company uses EKS and we just send dockerfiles and yaml files at it. We don't ever actually interface with an ssh prompt on our servers
I think Proxy _is_ overkill and perhaps expensive for what it is. For a pg install, running pgBouncer isn’t difficult. But I guess that depends on whether you’re authenticating with IAMs, which doesn’t sound trivial with bouncer (it might be).