Unfortunately a USB-C to USB-A Adapter won't work because Apple changed how Target Disk Mode works. If one of the Macs is running macOS 11.x+, then Target Disk Mode will only work using the Thunderbolt protocol.
from: Transfer files between two Mac computers using target disk mode - Apple Support
Note: If either of the computers has macOS 11 or later installed, you must connect the two computers using a Thunderbolt cable.
In that case the OP would need an Apple USB-C to Thunderbolt 2 Adapter and an older style Thunderbolt cable with the mini-Display Port connectors.
Also, Target Disk Mode is not an option during initial setup with Setup Assistant since there is no way of selecting the older Mac with Setup Assistant. I found this out while trying to migrate my new Mac using Target Disk Mode. It should be possible to use Migration Assistant with Target Disk Mode if the OP first creates a macOS user account on the new Mac and launching Migration Assistant.
Plus if the old laptop is using a slow spinning hard drive, the transfer will be painfully slow no matter what. If the older Mac is using a hard drive, then that hard drive is probably worn out or even failing which will make that drive even slower.
There is a possibility of using a USB to SATA Adapter if the OP want to remove the drive from the older laptop to try connecting it to the new Mac.