Do radiators always have to be placed under the windows in new buildings?

Do you run the radiator at the same temperature as the underfloor heating?

No, those are different circuits.

No offense, but then IT IS an evil. An expensive one.

For , it was selected, visual reasons for which one also paid.
With the OP, savings are hoped for.
Also the wrong approach for me nowadays. However, the reason for the hoped-for savings is missing here. Many builders with concluded contracts are now trying to regulate the construction costs.
So if the OP already has to/wants to dawdle in the planning phase, I would advise: plan properly first, then save in due time.

we are currently planning and trying to save costs and are considering replacing the FHB with regular radiators.

I would recommend looking for more suitable places to save. A new car with crank windows and no power steering is nowadays "below standard". You are planning/planned all sorts of things, what exactly is the current project: a new building by the carpentry company, if I understand correctly?
That would at least be a one-off "manufacturer" – with an industrial supplier I would strictly advise against it for that reason alone, because it would lead to the risk of "special quirks likely causing complications." Changing from the mainstream routine is "more dangerous" than what you can "save" in procurement. I believe even (who, like me, is known not to chase after modern gimmicks) has floor heating.

However, we do not want them in front of the windows... so we are asking ourselves: Do you still have to install radiators under the windows in a new timber frame house today, or is it just as good elsewhere in the room?

Elsewhere they usually stand just as stupidly. Radiators are from a time when heating niches were under the windows (which back then still regularly had waist-high parapets).

@Steffi33 also has radiators and has them super concealed

But in a room divider sculpture (see post #9) I would not compare that to mundane installation locations.

Thank you for your feedback. So there are indeed different opinions. We are reluctant to lay (have laid) a screed in any form. We are afraid that the FHB has a leak and something will have to be replaced in 10 years because of it.

Ok, now it's getting a bit wild. First costs and now no screed at all? Or no underfloor heating in the screed because of a possible leak? The pipes to the heating bodies are also not surface-mounted :)