Unfortunately you have been caught in a problem many have stumbled over. The makers and sellers of these devices for connecting several fans to a single mobo header are TERRIBLY INCONSISTENT with their labels! They confuse Splitters and Hubs badly!
I prefer to label them according to their electrical design and FUNCTION, and NOT by their physical appearance. So to me a SPLITTER is a very simple device that merely connects all of the Ground and Voltage Supply leads from all of the fans to the appropriate mobo header pins: specically, all Grounds are fed from Pin #1, and all +DCV power lines are fed from Pin #2. Next, note that any mobo header can accept the speed pulse signal coming back to its Pin #3 from only ONE fan, so all proper Splitters will send back only ONE fan's speed signal, and ignore all the others by simply NOT connecting to the Yellow lines from the other fans. Assuming you are using a 4-pn Splitter, the PWM signal from Pin #4 of the header also is connected in parallel to all of the fans' fourth lines so they all get the same PWM signal. Note that the only source of power to all the fans is from the mobo header Pin #2, so the 1.0 A max curent draw limit of the header applies to the total of all the fans connected to this header. Physically, a Splitter may look like a group of several arms of wires, or like a small circuit board with header pins. It has only TWO types of "arms": a single cable to plug into the mobo header, and two or more output "arms" (OR headers on a board) to plug fans into.
A HUB is a different device. It gets all of the +DCV power for all its fans directly from the PSU, and for this purpose it has a third "arm" that must plug into an output from the PSU - either a 4-pin Molex or a SATA power output. This gets a fixed +12 VDC supply that is distributed to Pin #2 of ALL the connected fans, and those fans do NOT draw ANY power from the mobo header, so the 1.0 A limit does not apply. The Hub does connect all its fans' Ground lines to header Pin #1, and all the fans' PWM signal lines to header Pin #4, but this deos not overload the header. And like a Splitter, the Hub sends back to its host header the speed signal of only ONE of its fans. A HUB may also appear to be just a collection or wiring "arms" such as you have, or it may look like a circuit board with header pins, or like a closed box with ports along the sides. It's pretty hard to telll a
SPLITTER and a HUB apart from simple appearance.
So you can see, the way you CAN tell the difference between the two is the presence of that third "arm" on a Hub that must connect to a PSU power output. OP, if you look at that thing you got, it HAS this power connection arm ending in a 4-pin Molex male connector (with only two pins installed in it), so it is really a HUB as I label them. A SPLITTER has no such third "arm".
Now, the problem this causes is this. The ONLY way to control the speed of a THREE-pin fan is to vary the Voltage supplied to it from mobo header Pin #2. That method of fan speed control is called "Voltage Control Mode" or "DC Mode". But a FOUR-pin fan works differently. It gets a constant full 12 VDC supplied to it from mobo header Pin #2, and then also gets the PWM signal from Pin #4. Inside that fan is a small chip that uses the PWM signal to modify the flow of current from Pin #2 though the windings, thus altering the motor speed. This method of fan speed control is called "PWM Mode". But if you plug a 3-pin fan into a 4-pin header using this PWM Mode, it gets always full 12 VDC from Pin #2 and never gets the PWM signal from Pin #4 because it has no special chip to use that signal, anyway! So a 3-pin fan mis-connected this way always runs full speed - you have NO control of its speed.
NOTE that a HUB (such as you have) sends full 12 VDC power to all its fans on Pin #2 and the PWM signal on Pin #4, and its design depends on assuming you have only 4-pin fans. So it (and its associated mobo header operating in PWM Mode) can NOT control a 3-pin fan's speed. The ONLY device you can use to connect several 3-pin fans to a mobo header is a SPLITTER that does NOT send a fixed 12 VDC from the PSU to its fans, But this DOES mean that all the fans draw their power from the mobo header's Pin #2 and cannot draw more than 1.0 A total. (Actually, you CAN use a 4-pin SPLITTER with 3-pin fans, because they don't make any use of Pin #4 anyway.) What is important, though, is that the HEADER MUST be configured to use the older Voltage Control Mode (aka DC Mode), and NOT the new PWM Mode.
Now, OP, you are concerned about that 1.0 A load limit. You have not told us what fans you have, nor how many. Many simple fans draw 0.1 to 0.25 A each max, so up to four of those can be connected safely to a single mobo header. (By the way, apparently the small starting current surge of the motors is taken into account in that 1.0 A max current draw spec.) So tell us what fan maker and exact model number of fan you have, and how many. Then we can recommend how to power AND control them from the TWO CHA_FAN headers you have available. It might be one Splitter or two, or even a particular unusual Hub that can deal properly with 3-pin fans.