OP, just to clarify labels first. Most fans now with lights in them have TWO cables. One ends in a smaller connector about 3/8" wide with three or fours holes and two small ridges down one side. The cable has either 3 wires (older fan types) or 4 wires. This fits on a mobo fan header with the two ridges sitting around a plastic "tongue" sticking up beside the pins of the male mobo header. The second cable ends in a wider connector that could have three or four holes, and the cable has a matching number of wires. The version with FOUR holes is for the plain RGB system. The one with THREE holes looks like it had four but one is blocked off, and it is for the Addressable RGB system (ARGB). This plugs into a mobo matching lighting header with pins. You cannot plug the 3-hole female connector into a 4-pin male plain RGB header. This is NECESSARY because the Voltage and signal systems used in these two types are quite different and they cannot be mixed on one circuit. As you have seen, you need something added to connect the ARGB lights in your fans.
Short answer, YES, you can and should buy a third-party ARGB Controller. There are about three major designs to choose from. All have a connection to a power output connector (often SATA) from the PSU for power. One is a box that has connection male output sockets and buttons on it to make settings MANUALLY for lighting display type and speed. You must be able to access those buttons, but usually this unit is in the wires you mount inside your case, so that can be awkward. The second has the Controller you mount in the case plus a separate box as a battery-powered remote control you use by hand from outside the case. Again, all settings are done manually using buttons on the box. The third type has the power and output connections, no buttons but also a third cable that must plug into a mobo USB2 header. It comes with free software you install and run on your system. You use that from keyboard and screen to make settings of the display. It is the most convenient design. One example of this latter type is the Razer Chroma ARGB Controller in the link provided above by stuff and nonsense.
Your other option is a Converter. This is a box that will accept the signals from a mobo plain RGB header (4-pin system) and convert that to safe signals and Voltage for a 3-pin ARGB lighting system. However, that plain RGB system cannot create any signals for the complex displays like moving rainbows that an ARGB system can do. So using such a Converter means you do NOT get ALL the complex displays of ARGB. Further, you need to be careful in selecting. Some of these have odd non-standard sockets to plug in your fans' lighting cables. Some are actually Controllers themselves witth manual buttons on the box that you must be able to reach to make adjustments. They have the added feature that you CAN connect a cable to a mobo plain RGB (4-pin) header and they will convert those signals to output ARGB signals on one or more ports. However, their default setting is to use the manual buttons. To turn over control to the input from a mobo header you must hold down one of the box's keys for several seconds, but the box does NOT remember that setting. So every time you reboot, you neeed to do the hold-down move to get mobo control.