I managed to solve the problem. The issue is that the new MBPs aren't exactly "standard" in how they handle the USB traffic. Adding a USB hub (USB-C Digital AV Multiport Adapter in my case) in-front of the switch solved the issue.
Turns out this was a known issue: https://apple.stackexchange.com/questions/342597/should-a-usb-switch-work-with-late-2018-macbook-air/410444#410444
This is what my `ioreg -p IOUSB` now looks like:
+-o Root <class IORegistryEntry, id 0x100000100, retain 23>
+-o AppleUSBXHCI Root Hub Simulation@00000000 <class AppleUSBRootHubDevice, id 0x100000405, registered, matched, active, busy 0 (0 ms), retain 13>
| +-o USB 10/100/1000 LAN@00200000 <class AppleUSBDevice, id 0x100000407, registered, matched, active, busy 0 (0 ms), retain 14>
| +-o USB3.1 Hub@00100000 <class AppleUSBDevice, id 0x1000019d1, registered, matched, active, busy 0 (0 ms), retain 12>
+-o AppleUSBXHCI Root Hub Simulation@01000000 <class AppleUSBRootHubDevice, id 0x100000418, registered, matched, active, busy 0 (0 ms), retain 8>
+-o AppleUSBVHCIBCE Root Hub Simulation@80000000 <class AppleUSBRootHubDevice, id 0x100000433, registered, matched, active, busy 0 (0 ms), retain 15>
| +-o FaceTime HD Camera (Built-in)@80200000 <class AppleUSBDevice, id 0x100000435, registered, matched, active, busy 0 (0 ms), retain 14>
| +-o Apple T2 Controller@80100000 <class AppleUSBDevice, id 0x100000439, registered, matched, active, busy 0 (0 ms), retain 13>
| +-o Apple Internal Keyboard / Trackpad@80500000 <class AppleUSBDevice, id 0x100000440, registered, matched, active, busy 0 (1 ms), retain 20>
| +-o Headset@80400000 <class AppleUSBDevice, id 0x100000446, registered, matched, active, busy 0 (0 ms), retain 11>
| +-o Touch Bar Display@80600000 <class AppleUSBDevice, id 0x10000044a, registered, matched, active, busy 0 (0 ms), retain 13>
| +-o Ambient Light Sensor@80300000 <class AppleUSBDevice, id 0x10000044e, registered, matched, active, busy 0 (0 ms), retain 11>
| +-o Touch Bar Backlight@80700000 <class AppleUSBDevice, id 0x100000456, registered, matched, active, busy 0 (0 ms), retain 11>
+-o AppleUSBXHCI Root Hub Simulation@14000000 <class AppleUSBRootHubDevice, id 0x10000046a, registered, matched, active, busy 0 (0 ms), retain 9>
+-o USB2.0 Hub@14100000 <class AppleUSBDevice, id 0x1000019be, registered, matched, active, busy 0 (0 ms), retain 14>
+-o USB-C Digital AV Multiport Adapter@14120000 <class AppleUSBDevice, id 0x1000019e2, registered, matched, active, busy 0 (3 ms), retain 12>
+-o USB2.0 Hub@14110000 <class AppleUSBDevice, id 0x100002349, registered, matched, active, busy 0 (1 ms), retain 14>
+-o IOUSBHostDevice@14114000 <class AppleUSBDevice, id 0x100002364, registered, matched, active, busy 0 (1 ms), retain 14>
| +-o HHKB Professional@14114100 <class AppleUSBDevice, id 0x100002378, registered, matched, active, busy 0 (3 ms), retain 12>
| +-o USB Receiver@14114300 <class AppleUSBDevice, id 0x10000238b, registered, matched, active, busy 0 (6 ms), retain 16>
+-o USB2.0 Hub@14111000 <class AppleUSBDevice, id 0x1000023ed, registered, matched, active, busy 0 (1 ms), retain 15>
+-o Samson GoMic@14111100 <class AppleUSBDevice, id 0x100002405, registered, matched, active, busy 0 (11 ms), retain 18>
+-o HD Pro Webcam C920@14111200 <class AppleUSBDevice, id 0x100002454, registered, matched, active, busy 0 (13 ms), retain 19>
+-o Flash Drive@14111300 <class AppleUSBDevice, id 0x100002f5a, registered, matched, active, busy 0 (4 ms), retain 12>