I doubt you are dealing with a clog. On several occasions the pump did not start by itself, but a small mechanical movement of tubing caused it to start and keep running. That may be a faulty pump, or you may have made an error in its power connections.
Let's try checking the power idea first. From what I can see on-line this system comes with a PUMP that has two cables - one ending in a 3-pin standard fan connector, and another ending in a wider 4-pin plain RGB connector. Similarly the rad fans come with two cables each, one for fan motor and one for plain RGB lights. It also comes with a fan Splitter to let you connect both fans to a single mobo header, and with a RGB Splitter, I believe, to connect all the lighting cables to your mobo's 4-pin plain RGB header. From the cooling perspective, the design is that the pump should always have a full 12 VDC power supply so it always runs full speed, and the rad FANS are the items whose speeds are adjusted automatically by their mobo header.
You have not told us what mobo you have, and that would help us give more precise advice. What is the maker and exact model number of your mobo? Also tell us exactly which mobo fan headers you have plugged in the PUMP unit and the rad FANS with Splitter.
The motor cable from the PUMP is a 3-pin one, I believe - it has 3 holes in it. Right? It needs to be connected to the CPU_FAN header ideally so that its speed signal can be monitored for failure. If you do that, then the CPU_FAN header needs to be configured to use the new PWM Mode, and not the older Voltage Control Mode or DC Mode. This trick uses a quirk of the fan type mis-match - when you plug an older 3-pin fan (or pump) into a header using PWM Mode, that fan (or pump) will always run full speed, which is exactly what you need.
That leaves us to figure out where to plug in the rad fans, which ideally also should be plugged into that same CPU_FAN header, since that header uses the temperature sensor inside the CPU chip to guide its control system. There is a way to do that, but some mobos also have an additional header called something like CPU_OPT which does use that same sensor and can be used for these rad fans. So what your mobo details are affects exactly how to do this. Another important factor we need to know is the FAN motor connector on the two rad fans - does it end in a connector with 3 holes or 4?
Post back with answers to those questions and we can complete the advice for how it should be connected. After than we'll know whether that solves it, or whether you may have a faulty pump.