ᐅ Heating System Choice: Air-to-Air or Air-to-Water Heat Pump

Hello everyone,
we are just at the very beginning of our home building journey – specifically in choosing the construction company. Neither of us are experts, and we are currently educating ourselves a lot -> please excuse us if some technical terms are unfamiliar or used incorrectly 😉
Among many other questions, we are uncertain about the heating system.

Construction Company A strongly recommends an air-to-air heat pump – meaning a central ventilation system with 'heating through the air flow.'
Company B suggests an air-to-water heat pump – combined with underfloor heating + still a central ventilation system.

And now about our lack of knowledge…
The air-to-air solution seems initially cheaper – but according to the advisor from Company B, there are disadvantages:
- Every room is the same temperature -> so having a cold bedroom and warm living room wouldn’t be possible?
- Warm air always rises -> so the floor would feel cold.

Neither of these issues would occur with underfloor heating since it can be controlled separately, and of course, your feet would be warm.
I don’t want to start a (new) fundamental debate – just wondering if you see it the same way? Unfortunately, we don’t know anyone who heats ‘with air’ 😉