No, it doesn't work like that. We are talking about just pure data, the adapter doesn't know what type of data is being transmitted.
If I plug in a USB C pendrive in it, it will transfer data because it is pen drive data but what if I have an MP3 or MP4 file on the pen drive. Will it stop transferring saying that whoa dude I can not do audio or video.
I agree with your point that it is most likely USB2.0 because it you are not getting USB3.1 at ₹150. But it is still USB2.0, right? It will transfer data at USB2.0 speeds. A DAC doesn't need to transfer data at USB3.1 speeds. It wont even utilise 5% (just an approximate guess figure) of USB2.0 speed.
You might be confusing Video Over USB C.
This might be helpful. USB C is just a physical connector, just a shape, it supports many different USB standards and protocols but they all are DIGITAL.
USB-A supporting USB3 standard has 9 pins and USBC has 24. How can you expect any adapter to ideally map all the USB-C standard and protocols to USB-A. There is nothing like ideal USB-A to USB-C adapter.
USB-C can support Thunderbolt while USB-A is physically incapable of it.
USB-C has 4 pins for power and 4 for ground. It can transmit huge amount of power while USA-A has one (USB3 can do max 900mA at 5V, USB2 can do max 500mA at 5V). Simply put, USB-C can not be fully adapted into USB-A connector because USB-C is far too advanced.