Alternative to Proxon air-to-air heat pump?

With the air heating system, you cannot spontaneously heat individual rooms separately, as far as I know.

Quick heating is not possible with a heat pump; it has far too little power when properly planned. It takes days to get warm. Also with gas: new buildings are too slow to cool down quickly.

Build with wood/low mass in the house, large gas heating and windows permanently open. Then you can keep old habits, but you will have heating costs like in old buildings.

The underfloor heating does not get warm (<30°), that is old thinking. However, it should be planned as well as possible and be on around the clock, in every room. The longer (time-wise) and larger the area, the cooler it can be operated and the less electricity the heat pump requires.

The opinions apparently turn quite strongly against air/air as far as I can see here. Thanks for that first of all.

Okay, I can probably also get used to underfloor heating, the only thing that might bother me I think is that it might be constantly too warm.
I currently live in a timber-framed house and I'm the type who's always got windows open. If I'm too cold, windows closed, radiator a bit on, then it gets warm. Otherwise, my windows are always open and I heat relatively spontaneously and only when I need it. Probably the classic radiator would still be the best choice for me, but who still wants radiators in a modern new building.
I hope you can understand my problem and why I considered an air/air heating system.

No, an old building would be optimal for you.

In modern houses, a temperature change takes about 24 hours. You set your underfloor heating with a heat pump to a temperature once, then you have that in all rooms, 365 days a year.

Anything deviating from that is either hard to achieve (even with "completely off" in the pantry it still has 19-20 degrees—the insulation to the outside is too good, and heat comes through the interior walls from the other rooms) or energetically nonsense (flow temperature so high that the bathroom has 25 degrees and the other rooms are throttled down to 20 degrees) which is either affordable (gas) or financially ruinous (air-water heat pump).

The air-air heat pump is an invention for passive houses and should stay there. In the worst case, still an exhaust air air-air heat pump. Then the money pit is official.

It will be a Galileo-Home. House with a wooden core. Completely diffusion-open, meaning solid wooden wall, insulation and plaster on the outside. No foil, no vapor barrier or vapor retarder. Inside, the walls are also completely solid wood, covered with drywall or clay boards, depending on what the budget allows.

The home builder recommends a heat pump in combination with underfloor heating for this type of house, and if I read your opinions like this, I am slowly also leaning towards underfloor heating...

Yes, take underfloor heating, preferably with a brine-water heat pump, if not possible or not desired, then an air-water heat pump. I would not want an air-air heat pump either (except in a passive house)!

By the way: Whether foil or not does not matter at all, because by law houses must be 'airtight', this is regulated by the Energy Saving Ordinance. Everything else is marketing nonsense (vapor-permeable seems to appeal to many builders, ... )

Is not a classic air-to-air heat pump Installed with us. We have an air-water inverter pump in the system. It works like a classic air-water heat pump, e.g. LWZ 504, only the underfloor heating is missing Our system is quite sluggish – I suspect even more so than a modern underfloor heating system. The airflow cannot be tripled at once and the temperature raised to 40 degrees. Then it drafts and hot air comes out of every air outlet like a hairdryer. Probably the same with the Proxon spontaneous heating with heat pump and energy saving regulation house does not work. If you want it to get warmer quickly on the sofa in the evening, then wood stove or infrared. Allegedly, in the passive house sector more and more is being done with a low base temperature and short, targeted heating as needed.

I believe most people here have not understood the Proxon air heating system. It is not an air-to-air heat pump! It is a normal ventilation system with heat recovery through a cross-flow heat exchanger, which transfers up to 85% of the heat from the exhaust air to the supply air. The reheating (essentially the loss compensation) is provided by small electric heating elements in the ceiling outlets of the supply air in the rooms. This allows each room to be individually, quickly, and easily tempered. The necessary electricity should ideally be supplied by a photovoltaic system. A heat pump is not used here at all.
The statement that air is a poor medium for heat transfer cannot simply be left like that in this case either. Yes, air has a lower specific heat capacity than, for example, water, meaning I have to "move" more air than water to transport 1W. The ventilation system does this anyway, which is why air heat transport is calculated in cubic meters per hour and not in liters per hour. Secondly, the elements heat where the heat is needed, so there are virtually no transport losses.
The idea that a radiator or underfloor heating heats the room better is also nonsense. If a room requires 100W/h of heat, I have to bring these 100W there. No matter how.
The fact is, I feel the warm air directly on my body. The system therefore responds much faster than a water-based system.
A radiator first has to get warm itself and then transfer the heat from the heating water to the room air and surrounding surfaces for me to feel it, so it is slower than an air heating system.
Underfloor heating even more so because there is no convection component at all, and the entire screed must also be heated. Additionally, both water-based heating systems stir up dust in the house, whereas the air heating system filters it out.
The only heat pump part of the Proxon system is in the drinking water heating via an (exhaust) air heat pump. This is connected behind the exhaust air duct of the ventilation system. The air that leaves the house and has already transferred 85% of its heat to the fresh air in the cross-flow heat exchanger of the ventilation system has further heat extracted from it in order to use the energy and transfer it to the domestic hot water using heat pump technology. Doing it this way is actually very clever because instead of blowing the exhaust air outside at, for example, 10°C in winter, you also use the last bit of usable energy content.
Similar to a condensing boiler, where the exhaust gases, i.e., essentially the combustion waste product, are cooled down so much by a subsequent heat exchanger that almost all the energy is extracted from them and they cannot even rise up the chimney on their own but have to be actively vented, here another heat exchanger is connected "behind" the ventilation system.
Now you can also understand why Proxon systems often drip. Of course, this should not happen uncontrollably but via the drip water connection!
Warm air can bind more water vapor than cold air. So if I now lead the warm room air, which contains a relatively high amount of water vapor (from cooking, showering, sweating), over the cold incoming fresh air to recover the heat, condensation water MUST inevitably form! Therefore, filters, condensate pans, and condensate drains should be cleaned regularly.
From my point of view, such a system makes a lot of sense because it combines the ventilation system, which is already absolutely necessary today, with an inexpensive heating system, provided the entire system is properly coordinated (e.g., its own photovoltaic system with battery) and balanced.
Because as with a water-based heating system, ventilation/ventilation heating also requires a "hydraulic" balance. All supply air valves must be set appropriately for the room to ensure the exactly matching volumetric flow, just like heating valves. This is done using an electronic vane anemometer with a funnel, which is pushed over the respective connection.