Can a geothermal basket precondition the air for controlled residential ventilation?

Floor cooling of course does not replace an air conditioner. Overall comfort gain is ultimately only achieved by the latter.
However, floor cooling is indeed "effective" within its limited scope and still provides noticeable comfort improvement. At least that is our experience and that of all visitors who come in...

We also do not like cold floors, so we have parquet throughout the house (except the bathrooms).
The cooler wood usually feels pleasantly cool and is rarely too cold. After all, it is only a relative perception, because it should not actually get too cold, at least not like in your example.
Because the floor temperature should not be significantly colder than the room/air temperature by 5-7° to prevent condensation. So if my target room temperature is e.g. 23.5°C and the room heats up to 28°C, the floor temperature is about 21° and thus contributes to cooling. This of course only works with "continuous operation" and is preventive, so that the room does not heat up in the first place through automatic temperature control. Ultimately also only "limited", but we are quite satisfied with 23-25°C at current outside temperatures of > 30°C.

Yes, one usually connects a geothermal basket to a brine-water heat pump, right?

You are right, and yet there are (unfortunately few, but also) posts that operate it in conjunction with a controlled residential ventilation system. And here is where my question comes from... As I said, I have no idea about this at all...

Of course, underfloor cooling does not replace an air conditioner. The overall gain in comfort is ultimately only achieved through the latter.
However, underfloor cooling is indeed "effective" within its limited scope and thus still provides noticeable comfort improvement. At least that's our experience and that of all visitors who come in ....