That's true. Stateless apps are a lot easier.
I have myself had to put a stateful service in K8s even before they had StatefulSets and it wasn't a walk in the park.
However doing what you described is hard... anywhere.
Even if you do it all by hand it's going to be hard and most likely brittle. It might take a bit more time/effort to do that on Kubernetes but I would say you would end up being a better solution that can actually sustain change.
As stated before by many, Kubernetes is not a magic wand. But it does force you to build things in a way that can sustain change without a big overhead.
We all know that different tools have different purposes and I'm not advocating it's perfect by any means. All I'm trying to say is that this idea - that a lot of people have - that Kubernetes is 10x more complex than doing X by hand is an illusion.
All deployments I have seen so far have been immutable infi scale webapps. That is easy.