We're in the business of PostgreSQL support, and some of our customers use RDS for various reasons. Not having to care about the deployment is one of the usual goals of using a managed environment, but considering they subsequently go and buy support from a third party might be a sign of something.
Of course, my view is biased because we only hear about the issues - there might be a 100x more people using RDS without any issues, and we never hear about them.
In general, the pattern we see is that people start using RDS, and they're fairly happy because it allows them to build their product and RDS more or less works. Then they grow a bit over time, and something breaks.
Which brings me to two main RDS issues that we run into - lack of insight, and difficulty when migrating from RDS with minimum downtime. Once in a while we run into an issue where we wish we could connect over SSH and get some data from the OS, attach gdb to the backend, or something like that. Or using auto_explain. Not even mentioning custom C extensions that we use from time to time ...