Heating decision: air-to-air or air-to-water heat pump

Hello everyone,
we are still at the very beginning of the house construction - namely in the selection of the house construction company. Neither of us are experts and we are currently reading a lot to gain knowledge -> Please forgive me if some technical terms are unfamiliar or used incorrectly ;)
Among many other questions, the heating question is uncertain for us.

House Construction Company A definitely wants an air-to-air heat pump - so a central ventilation system with 'heating via the air flow'
Company B suggests an air-to-water heat pump - so with underfloor heating + still a central ventilation system

And now to our ignorance...
The air-to-air solution seems cheaper at first - but according to the advisor from Company B there are disadvantages:
- Every room is the same temperature -> bedroom cold and living room warm would then not be possible?
- Warm air always rises -> so it is cold at foot level

Neither of these issues would occur with underfloor heating, as you can control it separately and of course your feet would be warm.
I don't want to start a (renewed) fundamental debate now - just do you see it the same way? Unfortunately, we don't know anyone who 'heats with air ;) )

Google "Thermal comfort with radiant and convective heat"

Where does the energy come from in the air-to-air heat pump? From the exhaust air or the outside air? In my impression, usually only an exhaust air heat pump is installed here, which gives you a very efficient ventilation system. However, you have to compensate heat losses 1:1 with electricity. You should not expect too much from the floor temperature. Today, the surface temperature is usually around ~22 degrees. A really warm floor like in the 80s would lead to a sauna in the living room today. Likewise, you can only expect limited differences between the rooms. The exterior walls insulate significantly better than the interior walls, so that balances out a bit. Nevertheless, I think you can feel the difference between heated and unheated rooms when walking in socks. It is also possible to keep the bedroom 2-3 degrees cooler. But 25 degrees in living rooms and 17 degrees in the bedroom are not feasible. Basically, I would prefer a water-based system. This gives you full flexibility later on to replace the heat generator.

Where does the energy come from in the air-to-air heat pump?

It was mentioned to be from a crossflow heat exchanger – so from the exhaust air if I understand correctly.

So, in summary, with underfloor heating small differences in temperature are possible, with air it is equally warm everywhere?
It was also mentioned that you could close the air outlets if you don’t want to heat the room – but that would also worsen the air quality (mold formation) – right?

I don't want to start a (renewed) fundamental debate now - just do you see it the same way? Unfortunately, we don't know anyone who 'heats with air ;) )

You can search for Proxon here in the forum (also on the web). Air/air (normally exhaust air heat pump) is predestined for house construction that lies between KfW 40 and passive house. There, it is insulated so well that the heat demand is extremely low. With KfW 55, you will (in all likelihood) lose the saved investment in the first heating periods through the electric auxiliary heating of the air-air heat pump. Otherwise, as already written, it is also a somewhat different heat. Rooms nowadays are also always equally warm (although a 2-degree difference counts as equally warm for me).

Regarding the different rooms: With warm air heating, you can reduce the heating/cooling - but only by reducing the airflow. In my grandmother's house, we had a warm air heating system based on wood/coal. Of course, that was primarily a heater; the house ventilated itself automatically and uncontrollably ;)

The bedroom is a supply air room; if you want to keep it cool in winter, you would have to reduce the air supply, which I find rather unpleasant.

In general, with air-to-air systems, a clean separation into supply air rooms (bedroom, living room, children's room...) and exhaust air rooms (bathroom, toilet...) is not possible. I would find that very unpleasant.