External drive not seen with USB C, only USB C to USB to USB C

Hi,

Have a 2020 MacBook Pro with two Thunderbold 3 ports. It's running Catalina 10.15.7. Was given a 2 TB drive (secondhand, so don't have the specific specs on it, but it has a single USB C access port and no other inputs) to use as a Time Machine backup, but I'm running into problems.

I have a USB C to USB C connector (Amazon jargon calls it "Short USB C to USB C Cable, Lamtoon USB 3.2 Gen 2 Type C Cable 10Gbps Data Transfer 100W PD Fast Charge Cable FPC Design, 4K@60Hz Video Output, Thunderbolt 3 Compatible for MacBook Pro," if that helps), and my laptop does not recognize the drive when plugged in using it. No indicator light on the drive, no device showing up on the laptop. Curiously enough, the laptop-side of the cable gets warm after it's plugged in for a while, so there's... some kind of communication happening. The cable can get power across when plugged into my adapter fine, too, so I don't think it's just broken.

I tried resetting NVRAM and SMC and it's still dead. I know the drive itself works: the laptop sees it and can successfully use it as a Time Machine drive when I connect it USB C to USB to USB C, but (in addition to just feeling plain silly) this only gets a few MB across per minute, meaning it's next to useless for the 340 GB I'm trying to back up.

Is the drive itself just not compatible? Do I need a different cable? Is there anything else I'm missing here? Would hate to have to give up and just get a new drive.