This will work, in theory, and if you’re willing to use a lot of water. It’s probably a bad idea.

Heating one kilogram of water by one degree Celsius without phase transitions (freezing/melting, evaporating/condensing) takes 1 kilocalorie of energy. That’s roughly 4 kilojoules aka kilowattseconds, or 0.0012 kWh.

Thus, to get 1.2 kW of cooling, which is about half of what those tiny portable air conditioners promise, at a 10 degree temperature difference, you’d need 100 liters of water per hour. If water costs $0.40 per 100 liters, and electricity cost $0.40 per kWh, an air conditioner (using about 0.4 kW of electricity to pump 1.2 kW of heat) will be a lot cheaper, and that’s ignoring the power you might need to run the pumps and fan on your solution (all of which you get back as heat!)

Unless the water in the loop is below the dew point, you also won’t get any dehumidification. This is actually more important than cooling, and a big reason why air conditioned rooms feel so much better (sitting in the shade in 40° C dry weather would be unpleasant but fine, at 100% humidity, it would be reliably fatal regardless of fitness).

If you’re building new, look into:

• proper insulation
• insulation and windows that optimize for the right thing for your climate (in countries like Germany, I suspect windows are optimized to let as much heat in and as little out as possible, which saves heating costs in winter and turns apartments into hellholes in summer)
• passive cooling paint and panels - I don’t know if they’re commercially available and in a practically usable state yet.
• solar to power the AC
• swamp coolers aka evaporative cooling (the split kind that evaporates water outside). Downside is they use water (which actually is lost - evaporated), so if you’re in a drought prone area where water is restricted or expensive they might not be the best choice. Also, it has to be actually dry (low humidity) when it’s hot. Get actual, local climate data, not gut feeling. Check if there are commonly used commercial solutions, possibly combined with actual A/C (very common for industrial scale setup, not sure if common for home setups).
• regular air conditioning. I’m assuming you’re trying to build a house to live in, not an art. Economies of scale mean that going with suboptimal but standard solutions almost always beats custom hacks. If you have the same brand of AC as everyone around you, the repairman will know how to repair it, will have spares, will know how to design it so it is sufficient for your house, etc. - if you build something yourself, you will be the only one who can maintain it.
• ceiling panels - these cool the room by running cold water (generated using normal A/C heat pumps) through pipes/panels under the ceiling. The upside is that they also remove radiant heat, the room feels about two degrees colder than it is thanks to this (look up “wet bulb globe temperature” for a rabbit hole). The downside is that they can’t dehumidify and actually stop working in high humidity when you’d need them the most: if you run water colder than the dew point through them, it’d condense and start dripping all over your stuff, so it shuts down or limits how cold the water can be (and thus how much it cools). Consider them as an addition only if they’re common and installers are familiar with them.

In the end, you’re building a new building, so you now have a chance to do everything right using modern but already proven technology. I wouldn’t DIY anything critical and hard to change like this. Remember, you’re trying to find the best (likely: cheapest in the long term while meeting your reliability requirements) solution that will solve your problem. There’s a very high chance that’s simply “add more A/C and solar according to what’s locally available”. And that’s fine. There’s nothing bad about that.

I wouldn’t, for example, try to build with different materials than locally common, even if those were “better” by some metric. That often doesn’t give you a better house, that gives you a unique house, and unique can be a nightmare.

You’re describing a fancoil supplied with cool, regularly replaced, municipal water (normally this water would be a fully closed loop cooled with an air source or ground source heat pump). Your energy needs will just be a circulation pump. You’ll probably notice a little cooling but it depends on how cold the water is, the surface area of the radiator, and the flow rate of the water. It has the advantage of being low maintenance so give it a shot and perhaps build it in a way you can access the components and improve / experiment over time.

Look into an approach / methodology called Passive House. Passive House focuses on making buildings that have near zero heating and cooling load. If you get the math right / design from scratch with this in mind you can make a Passive House in nearly any climate. Common modern single-family-home building techniques are generally not at all closely aligned with building a Passive House.

When trying to keep a house cool, here are the things I would focus on (in order of priority):

1. Reduce solar heating impacts: either place shade trees or awnings to block direct sun on the entire structure (or the windows at a minimum).

2. Build a highly-insulating enclosure (~R30 walls and ~R50 roof at a minimum, but you could push that further). If you are set on building with lumber you still can, you could building an offset double-stud wall filled with insulation, and of course an appropriate amount of exterior insulation a well. The goal in addition to insulation quantity is to reduce thermal bridging. Consider a “simple” house layout. Avoid too many corners / details / flourishes that add construction complexity.

3. Utilize free-cooling first: as your first stage of cooling, open large windows close to the ground and open clearstory windows in the roof / top of a stairwell or similar, it really depends on the layout of the home (and ideally the layout is design around this concept). This allows the heat to be drawn out naturally via convection. Include ceiling fans for comfort. This approach will work until outdoor air temperatures get quite high. Once free-cooling will not longer work

Once free-cooling will not longer be effective you can transition to mechanical cooling. Close all windows and cool your space either a high-efficiency air-source heat pump (and / or your free-cooling municipal water fan coil).

4. Similar to the design methodology to encourage natural air / heat flow out clearstory windows or “solar chimneys”, also consider just having higher ceilings where heat can pool but you won’t feel it. Your exhaust should pull from these areas.

5. Dedicated outdoor air system (DOAS): don’t design your mechanical ventilation system to cool using air (aside from the free-cooling described earlier). It’s inefficient. Hydronic heating and cooling (moving heat with water) is much more efficient. That means heat pumps for heating as well as cooling. Mechanical ventilation rates should be the bare minimum, just enough for fresh air but not for temperature control. Perhaps look at flow rates included in ASHRAE 62.1 or a standard more focused on residential homes. Also, your supply air can be separately ducted to each room (not a shared trunk), each being much smaller than what you see in a “normal” house, this gives more control for every single room.

6. ERV: of course you’ll want to install an energy recovery ventilator to capture what heat / “cold” you’ve worked to produce before instead of throwing it away along with your exhaust air.

7. For heating your domestic water, get a heat pump hot water heater (with tank). Instead of making heat it takes heat from the surrounding room and puts it into your domestic water tank. That means it “outputs cold” into the surrounding room, the opposite of a gas or electric resistance water heater.

8. Earth tubes: to naturally pre-condition your supply air by running it through the ground first. Another form of free-cooling but useful when the house is “buttoned up” because outdoor air temperatures are too high. This is when you’re only supplying minimum ventilation air.

9. Limit the things in the house that make heat. Efficient refrigerators / freezers (see energy star website), computers that are no more powerful than what you need, etc. Place these things in areas where the heat won’t bug you as much.

Hope this helps.