You have not specified exactly which MSI X570 board you have, but I believe it has both a CPU_FAN1 and a PUMP_FAN1 header. The Cougar AIO system you have is designed so that the PUMP unit power cable should plug into your mobo PUMP_FAN1 header, and it will always run full speed. The fans on the radiator are to be connected (using the Splitter supplied) to the CPU_FAN1 header. IF you have only two fans and the Splitter has three outputs, be sure that ONE of those fans is plugged into the only Splitter output with all 4 pins. The speed of that one rad fan will be shown as the CPU Fan Speed, and the speed of the other will be ignored completely.
In BIOS Setup, each fan header is configured separately. It is not clear in your mobo manual what Profile options you have. For each of those two headers you should start with the Standard Profile (that's the "fan curve"). At the upper left of the screen (see p. 36) set both to use the PWM Mode. Now, the specs shown for those fans say their speed range is 600 to 1800 RPM. So check the fan curve shown for the CPU_FAN1 header that the rad fans are plugged into, at the lowest temperature and speed. The lowest speed set should be around 600, but maybe a bit lower if you wish. That is the one you might want to change if it is set too low. You can just drag it with your mouse cursor up or down. When you have made all the changes you want on each header, use Esc to get back to Main Menu, then F10 to get to the Exit Menu. There choose to SAVE and EXIT to save your settings and reboot. See if that gets the fans to start and stay running. If not, try again with a higher minimum speed in the fan curve.