If you compare the hourly cost for any instance on EC2 vs RDS, it's significantly more expensive for the managed solution (75%+ more), which is to be expected. I know people who roll their own for cost savings.
If you are architecting everything on AWS trying to avoid “lock-in” you’re going to move slower and pay more - the worse of both worlds.
Neither costs a fixed, single amount. Both are ranges that depend on the competence of the practitioner. For example, it's cheaper to use a reserved EC2 instance for 1 year than just on demand. Similarly, the default cost for DIY is much higher than the lower bound of the range.
If DIY includes "enterprise" hardware, software, and/or support contracts (which arguably contradicts the "Y" in DIY), then the top end of that cost range is easily above the top end of the AWS cost range. In this way, those people are correct.
However, if we limit it to truly doing it yourself [1] and with commodity hardware, software, and methods, that's no longer true.
More importantly, as the GP mentioned, "If you're going for cost savings", it's the bottom end of the cost ranges that are key, and, even at modest scale, AWS is way above DIY. In this way, those people are incorrect.
[1] There's a limit, of course. For me, that limit is commodity versus custom. It's safe to outsource server manufacturing (all the way up through assembly, initial burn-in, and even rack/stack), since, it's easy enough to replace with a different vendor. It's not safe to outsource specifying what goes into those servers or final "smoke" testing. Sometimes, even for something that's safe to outsource, like remote-hands replacement of failed parts, it may not be worth the price.