Did the laptop cooler have a USB-A port as a means to supply power to it? Did the laptop cooler come with a cable that had male USB-A connectors on both ends to supply power? If so then you should be thankful no damage was done to any of your hardware then destroy this laptop cooler and it's power cable as it is a fire hazard.
Allow me to explain...
USB is a port that is used to supply power to devices as well as exchange data with them. This power flow specification is such that power is to only flow out of a USB female port (like that on a computer) and into a USB male plug. Violating this is like having a computer that instead of a standard three pin IEC C14 connector like most computers it had a NEMA 5-15R, like an outlet on a wall would. This is using a connector designed only for power output for power input. To make this work they had to include a cable with a standard electrical plug, like one would see on any home appliance, on both ends. Someone might not realize what this cord is for and plug in two outlets together. In many cases nothing happens because both outlets would be fed off the same electrical phase. In some cases this would be short circuiting a 240 volt AC supply capable of 15 amps before the circuit breaker tripped. This short circuit would mean a lot of current flowing, admittedly in a short time if the circuit breakers worked like they should, but not without doing some damage to the cord and at least one outlet.
The cord that came with this cooling pad is doing much the same thing. It would potentially short circuit two power output ports capable of supplying potentially 20 amps. There will be circuit breakers but they are not likely to stop the current flow fast enough to prevent damage. This cable needs to go. Because the cooling pad would only work with unsafe cables then it too must go.
You are not likely to find a cable with male USB-A on one end and a male USB-C on the other end to work in supplying power to this cooling pad. For power to flow from a USB-C port into a cable with a male USB-A connector on the other end would violate the USB specification. There's some very simple circuitry in the cable to prevent this.
During the development of the USB-PD version 1.0 spec there may have been a means to make power flow into a USB-A port safely. Maybe, most likely not, it's hard to find the spec any more since it was not widely adopted, possibly not adopted at all as I have seen nothing that supports USB-PD v1.0. USB-PD v2.0 and later quietly dropped USB-PD on USB-A ports, leaving USB-A ports as being specified to only supply power.
Maybe, possibly, the cooling pad and cable are doing things safely. Perhaps they built a cable that will not supply power until conditions are met and therefore supply power safely. That's not likely since a far less expensive means to get a safe connection was to use USB-B, USB-C, or some other connector.
If they cheaped out in such an obvious way, in blatant violation of a very important specification on safety, they likely violated the spec in ways that are not so obvious. Perhaps by taking far more power from the USB host than is allowed. Most computers are able to hold up to power draw beyond the spec without damage. That's playing with fire, perhaps literally, and could damage hardware.
I made this long explanation to hopefully make it clear on what the problem is and that the best course of action I see is to destroy this cable and cooling pad. It's quite possible I am making the wrong assumptions and your cooling pad is adhering to the spec. If that were so then it should work with any cable that also adheres to the spec. It's also possible is was built to the USB spec but is now broken. If you have a broken laptop cooler then we get back to tossing it out as a solution. Perhaps it can be repaired but that is not likely worth the trouble as I suspect this cooling pad was not a large investment.
For a cable with male USB-A on one end and male USB-C on the other to supply power to anything, as in the spec, then the USB-A end must be connected to the laptop and the USB-C end to the laptop cooler. From your description that does not appear to be the case.
Please clarify what ports are on the cooling pad, if it is as I suspect then the cooling pad was made broken and the only fix is to toss it out and replace it with a device that meets the spec. If this cooling pad is safely made then consider my answer a warning to all on paying attention to the ports on your USB devices. There are far too many that don't adhere to the spec. It's silly to try to save $10 on a USB device when it could damage a far more expensive laptop because the cables were not up to spec.
