It's actually a fine solution and I often do this with infrastructures that I don't intend to maintain myself. In general, however, I try to minimize complexity, and having an external service feels a tad more complex than running the database inside the instance. But you could probably make a successful case to the contrary - namely, that using a service rather than running your own is actually much simpler and easier to maintain.

I must confess that in this decision, the complexity of the AWS console (which I consider awash in buckets of accidental complexity) usually tilts my balance towards self-hosting.

AWS RDS requires a certain scale to make sense, it gets expensive fast, but the cost doesn't go up after the first bump.

If you don't need the features (updates, slaves etc) you can get away with just a normal DB in an EC2 instance.