First of all I would double check if the headphone jack to casette converter works. If it does then unfortunately it is likely that the splitter you bought just is not compatible for your device. Try shopping for a "USB Type C to 3.5mm Female Headphone Jack Adapter" specifically for your phone (under $10).
Just tested one of these adapters for my old Mi A2. The adapter is marketed specifically for Xiaomi Mi A2 and cost ~$5. It works also after plugging an USB-C to USB-C extender chord in between, sadly I have no USB-C hubs here. But the same device does do nothing when plugged into my Pixel 3a.
The apparent reason for this incompatibility is apparently lack of standardization according to Lifehacker.
The big problem with USB-C dongles is that nobody codified new hardware standards before smartphones started dropping their headphones jacks.
Playing digital media through headphones requires a digital-to-analog converter (DAC)—which changes digital audio data into analog signals your headphones can playback—and an amp. The disagreement is whether this conversion should happen on the smartphone itself or within the adapter you’re connecting to it.
By the way, if you are trying to simultaneously charge the device while using the USB-C audio dongle, I'm not sure that will work either, I would definateley get the pure dongle first and then maybe try it behind an USB-C hub with a PD (Power Delivery) input jack for charging for a long drive when I had to charge the device.