Yeah, it did appear to me that the connections at the top of that board are soldered. What I did NOT expect is that three of those wires go "to the mobo" and are obviously used to provide power and control the the three case vent fans plugged into the board's FAN headers. That is a board design I have never seen before. From what you describe, the board's fan MOTOR system apparently IS a basic SPLITTER, and not a Hub. The fans work only when the cable to the board is plugged into the mobo CHA_FAN 3-pin header, so those fans get NO power from any other source. Further, the CHA_FAN header cannot send out PWM Mode signals because it has no Pin #4. But I DO expect that those fans connected to that board have their speeds controlled by the CHA_FAN header, just as they would be if you were using s separate Splitter to make that connection. Is that correct?
Can you tell where the other wires from that board top go - to what mobo headers, or perhaps to a SATA power output connector from the PSU?
For the connector on the end of the cable from the pump unit, you are probably right. Re-examining the photo maybe I do see 4 wires. Certainly the AIO system web page specs say the pump has a 4-pin connector (well, we know that!). Those specs do NOT tell us whether the pump speed is always full speed or not - no speed spec at all. So, both possibilities exist. The pump MAY be wired as a new 4-pin fan with speed controlled by the PWM signal on Pin #4 of the header - that would be unusual, but possible. OR the pump may be wired just like a 3-pin fan as most are. IF that is the case, then as long as the pump is plugged into a header (like your CPU_FAN header) that is sending out signals in the 4-pin PWM Mode, it will always run full speed. For your particular situation - you have only ONE header (the CPU_FAN header) to use for the AIO system - you still should use a simple 4-pin SPLITTER to connect the pump AND both rad fans to that header. Again, ensure that the PUMP cable is plugged into the only Splitter output with all 4 pins. The pump will operate as the designers intended with its speed shown and monitored for failure. The rad fan speeds will be adjusted by the header, although they will not be shown anywhere.
Generally pump speeds are NOT intended to be less that full speed, but there ARE ways to do that using configuration options if the pump is the only device on one header. And there are a few AIO systems designed to alter pump speeds, but almost all of them do that with their own propreitary software tool. FEW people find that pump noise is troublesome, but some do. More often IF the pump makes noticeable noise that is because of a different problem - air bubbles trapped in the pump.
FYI, here is why it is NOT normal to have BOTH pump speed and rad fan speeds controlled in the same way. Those two devices have different response times - that is, the time from changing the speed to actually seeing a change in temperture is different. So suppose BOTH are being altered in the same way (from the same signal). Now the temperture at the sensor rises, so both speed up to remove more heat. The result will be that the measured temp drops. But it will drop quickly because of ONE of those changes and get really close to what is needed. Then a little later it will drop even more and get too cool. So now the system will slow BOTH down and try to fix that. Again, the system will adjust then overshoot. Repeat, repeat, etc. The solution is to have only ONE item (the rad fans) altered automatically while the other is kept constant - usually, the pump at max speed. In systems that allow it, the fan speed may be less than max but still fixed. This allows good cooling with slightly higher fan speeds, but it can limit the max amount of cooling available in high workload situations. In your case you can understand that if the rad fans AND the pump BOTH were running at 20% to 30%, cooling would be VERY limited and most likely BOTH would speed up to compensate, leading to that cycle of pump and fans "chasing" each other.
If you were to buy an extra 4-pin PWM fan and, using a 4-output Splitter connect that plus the AIO pump and 2 rad fans all to the CPU_FAN header, then yes, that extra fan would run under the same control as the rad fans. Because the extra one is not identical to the rad fans its speed might be somewhat different, but that does not matter. The actual speed is not important.HOWEVER, you must bear in mind the current draw calculation in Item 4 of my earlier post of April 17. That showed AIO system load to be 0.72 A max. So the added fan cannot have a max load over 0.25 A.
However, I am curious why you would do this that way. First of all, do you NOT have the three 3-pin case vent fans running under speed control through that odd board? Do they really run full speed all the time, or did that happen only when you connected them somewhere else? My thinking here is that you COULD connect a new 3-pin fan (pus one existing fan) to one output port of that same board using a simple Splitter. Although a 4-pin fan also could be used there (because, confusingly, the 3-pin Voltage Control Mode signal system can control the speed of a 4-pin fan), a 3-pin unit is likely a little cheaper. And I expect that board (and certainly the CHA_FAN mobo header supplying it) can handle four simple 3-pin fans - the existing 3 total only 0.48 A. By the way, the fact that the GPU board fan DOES come on and off at idle is NOT a problem, nor a symptom of inadequate cooling.
If you do replace your mobo, create a new thread for advice on how to use its features to best advantage. If you like, left-click on one of my message icons and send me a Direct Message to call attention to that new thread.
On your last item I can see two options. IF the fan power source you will use DOES change its fan speed by varying the Voltage output (the normal 3-pin Voltage Control Mode system), then some way to reduce that voltage in-line would give you both ALL speeds reduced and speeds varied by the header supplying that power. BUT that might NOT work as you expect. Remember that the header sends out whatever fan signal is required to get the actual TEMPERATURE at the relevant sensor to its target. So if you deliberately slow the fans down the temp will rise, and the header will demand more fan speed to offset what your Voltage Reducer device did! The net result would be only that you limit the MAX speed (and cooling) available.
On the other hand, if your issue is that the existing power supply system to your 3-pin fans can NOT alter the fan speeds and they always run full speed, then what you really want is a fan power supply system with MANUAL speed control. This is one example of that.
https://www.amazon.com/Cooling-YACSEJAO-Channels-1Knob-Controller(1Knob/dp/B0B2ZJK282/ref=sr_1_8?crid=32U0BJKBTCLLM&dib=eyJ2IjoiMSJ9.lTUimjtsmeNK-JIQF1CCjJ7AggDTXeO4dIgo2U4EP424nENS3wuBrW2qZpYSvYeKKLXAhy-TaqHe9hg78MtrvJA-aH5L5K_y9Xz51piQq_8UDJYIPt6V3ddIbhk9gdrj7DdteuS4AqVHqm7CGtTXC3AvcnKCmJ3t2D2WYyCDNGzYW33L0Bgw0gBRTrhWKityzWr-YOoSumlIPhsNZb3LfkdZJUxhG2neDhcPVdCpTTw.QyDy-4X2klqIsCHF-3d0vq8ky3ipwW2ABMyVrhn5WEM&dib_tag=se&keywords=computer+fan+speed+control+3-pin&qid=1745077977&sprefix=computer+fan+speed+control+3-pin,aps,126&sr=8-8
Its main plate mounts on the back of your case in an unused PCIe card slot. It gets all power from the 12 VDC lines of a 4-pin Molex power output from the PSU. It has three output connectors for fans that WILL accept either 3- or 4-pin fan connectors. What it really does is send to all its fans ONLY a VOLTAGE power supply that can be altered by its control knob. It does not try to use PWM speed technology nor can its send any fan speed signal anywhere. In your case, you would connect all your case vent fans to this controller and NOT use the odd board's fan motor outputs.