Eh, sort of. The difference is that the cloud can go find other workloads to fill the trough from off peak load. They won’t pay as much as peak load does, but it helps offset the cost of maintaining peak capacity. Your personal big server likely can’t find paying workloads for your troughs.
I also have recently come to the opposite conclusion for my personal home setup. I run a number of services on my home network (media streaming, email, a few personal websites and games I have written, my frigate NVR, etc). I had been thinking about building out a big server for expansion, but after looking into the costs I bought 3 mini pcs instead. They are remarkably powerful for their cost and size, and I am able to spread them around my house to minimize footprint and heat. I just added them all to my home Kubernetes cluster, and now I have capacity and the ability to take nodes down for maintenance and updates. I don’t have to worry about hardware failures as much. I don’t have a giant server heating up one part of my house.
It has been great.