Theoretically you could indeed drive cool water to the radiators to cool the house, but you don't get the convection effect that a radiator produces when hot, nor thermal radiation. So you need to rely on transfer of heat from the air to the metal of the radiator, potentially with no airflow. Easier to open the windows. In the last 12 years, I've never wished I had a system to chill the air in my dwelling; Ireland is generally on the cool side (though this may change).
This would be kind of running a mini-split AC to the rooms using water as the refrigerant instead of what hvac normally runs.
It's not a "this is the best choice" or even "this is a better choice than opening the windows" but rather a "if you did this, would it work? kind of?"
Yes, but it isn't very common at all for one simple reason: condensation. You have to keep the temperature of the coolant above the dew point, which requires knowledge of ambient temperature and humidity. That means your coolant isn't actually that cold, so it can absorb less heat before needing to be re-chilled. It's overall pretty inefficient.
In that case, yea, it would be a {something} to water heat pump to heat up the water, and then circulate that water through the house.
It would be curious if there were installations that did the other way too. In theory, you could also circulate cold water in the summer. The problem would have been "how do you make the water cold?" and in many older cases, that would be impractical. Though... that could lead to condensation on the radiators in the summer which could have downstream effects (condensation drips water on the floor).