Retrofit underfloor heating??

Ok, that still sounds quite reasonable...

My father has now found out that you can also rent milling machines that can mill recesses directly into the screed for the pipes.
Have you heard of that before?

I wouldn't do that for the simple reason that then the insulation under the underfloor heating is missing. You simply lose too much heat that way. In the dry system, you at least have 25 mm of insulation within the 45 mm build-up height, which isn't huge, but at least it's something. If the build-up height is lacking, you can still use a thin-layer system, which only has a 10 mm build-up height and is still better than milling out since old tiles or similar do not need to be removed.

And what about the milling method on the ground floor? The base slab shouldn't emit that much heat.

No idea if the floor slab is insulated, it wasn't like that for us. If you really want to mill, you can do that. I don't quite see the point. I prefer a bit more insulation and less dirt and effort, but to each their own.