The item you linked really is the common "4-pin fan hub" I described. It just does not have a central block of connectors. It gets fan power from one 4-pin Molex male connector that picks up only the Ground and 12VDC lines from the PSU. It uses one 4-pin fan female connector to a single mobo SYS_FAN port to connect to the fan speed (Pin #3) and PWM Signal (Pin #4). It has three output connectors of the 4-pin male design, but only one of those has all four pins. The other two, which I think you called 3-pin connectors, are just missing their Pin #3. This is to ensure that, of the three 4-pin (PWM) fans you connect, only ONE will actually send its speed signal back to the mobo port, and that is the correct way to do this. Each fan generates a speed pulse signal (2 pulses per revolution) and sends it back to the mobo on Pin #3 for counting. If you connect more than one fan to a mobo fan port, you must NOT send to the port more that one fan's speed pulse signal. Otherwise the port's pulse counting solftware would be very confused with many pulses at random timings.
So, that splitter is for use ONLY with 4-pin fans and a 4-pin mobo header, BOTH truly operating in PWM Mode.You must understand that what it does is send to its three fans a constant 12 VDC power supply on Pins 1 and 2, and a PWM signal on Pin 4. If you connect a 3-pin fan to an output, that fan will not receive the PWM signal and could not use it anyway, so it will always tun at full speed. Another way this plan can "fail" is if the mobo SYS_FAN port is really a 3-pin port operating in Voltage Control Mode. In this situation the varying voltage on mobo port Pin #2 is NOT passed on to the fans because the fans' power is coming from the PSU source via the Molex connector. Even if the mobo port has 4 pins, there are two possible ways for this to not work. One is that SOME mobos allow you to set the way the 4-pin port operates, either as PWM Mode or Voltage Control Mode. If you do NOT set it to PWM Mode, there will be no PWM signal to pick up and share with the three fans. The other possibility for "failure" is that SOME mobos provide "fake" 4-pin mobo SYS_FAN ports that really are only 3-pin ports with a fourth useless pin, and that operate only in Voltage Control Mode and fail to provide the PWM signal required.
OP, you hope was to use automatic control by one mobo SYS_FAN port of several fans via this splitter / hub. However, that cannot work for 3-pin fans that cannot use the shared PWM signal. Using this unit, all your 3-pin fans WILL work with proper power from the PSU (not overloading the single mobo fan port) BUT they will not be under control - they will all run full speed all the time. You could get that result without this splitter by merely connecting all the fans via adapters directly to a PSU 4-pin Molex output.
For your situation, you CAN control multiple 3-pin (Voltage Control Mode) fans from ONE mobo SYS_FAN 4-pin header IF you buy and use the Phateks PWM Hub I mentioned. Instead of simply sharing the PWM signal to its fans (as the splitter you linked does) the Phanteks unit uses that PWM signal to create six of its own 3-pin fan ports that do operate in Voltage Control Mode to run your 3-pin fans under proper control.
Hypothetically you could just connect all the 3-pin fans to a single PSU 4-pin Molex output using only the Ground and +12VDC lines, and inserting into the voltage line a variable resistor (wired as a series resistor, and not a true potentiometer). You'd need to do a bunch of calculations to figure out the proper resistance and the power rating of that resistor. You also would need to figure out how to deal with two significant issues for such a circuit: the fans cannot reliably start up from stop unless their voltage is at least 7 VDC, and at some slightly lower voltage (maybe 5 to 6 VDC) even a running fan with good bearings will stall and not re-start. Properly-designed third-party "fan controller" modules take care of these issues for you by designing circuits containing much more that a mere series variable resistor. Then you still have to understand that any manually-controlled fan speed control system makes YOU the "brains" of the control. That is, YOU must decide what speed the fans need to be running to keep your system cooled properly, AND you must monitor your system and change the fan settings as your workload changes. If you want your fans to be automatically controlled according to actual temperatures measured by properly-placed sensors calibrated for your system, that can't be done with third-party fan controllers. It CAN be done by the mobo automatic controllers IF you use the correct tools to connect fans to the mobo ports.