Rusty heating water with discolorations discovered.

I find it strange that neither a magnetic sludge separator nor a microbubble separator was installed in our system.

This is by no means strange but common practice. If not ordered, it will not be installed. Not always necessary.

And I wonder why a regular steel towel radiator was installed and normal C-steel pipes used from the distributor box of the underfloor heating on the upper floor to the towel radiator in the bathroom. Is that botched work?

No, that is not botched work, but also not fine engineering. As has already been advised to you, flush the circuits, fill them with demineralized water, and if possible, add a corrosion inhibitor to the system.

The steel pipes and the radiator are of course rubbish, but conform to standards. Here you can/must take action yourself and possibly replace everything with more noble materials.

Also, take a look at the expansion vessel. It is often made of simple steel as well.

And who is supposed to pay for that? We do not have the money to have the floor ripped out and the pipes replaced. That's at least 10,000 euros if not more.

And who is supposed to pay for that?

We don't have the money to tear out the floor and have the pipes replaced. That's at least 10,000 euros if not more.

Why? Flushing is done via the heating circuit distributor, the dirt separator is installed somewhere in the technical area in an accessible section of the supply line or preferably the return line.

Flushing + refilling is 1 day of work + a few cubic meters of water.
Dirt separator around 100€ + 1 hour installation
Plus travel costs. That amounts to roughly 500€-600€ + tax

Often a "normal" expansion vessel is used instead of one for drinking water; the pressure is then simply adjusted to a different level and it is rust-free. That's how I have the [Inst.].

Adding any substances to the heating water should also be done with caution. The danger is that the stuff can damage the heat exchanger. Have experienced it as well.

Mycraft wrote that it makes sense to have the pipes replaced, which is why I wrote about the minimum 10,000 euros.

If we now flush, refill, install a sludge separator and have regular maintenance done, will our heating system have a normal lifespan or do I have to assume that the pipes or the entire system will rust through after a few years?

The picture shows a pipe that was replaced after water damage (due to material or workmanship defects) because it started to rust from the outside. The building insurance is now investigating who is responsible.

Is it normal for a C steel pipe to look like this on the inside or should I be concerned?