While there is also potential for software conflicts like the EOS utility, and also google drive and other cloud drive software, this specific problem is rooted a bit deeper:

The camera can either charge via USB-C or transmit data via USB-C - but not both at the same time. When connecting, the camera has trouble deciding whether it wants to use the USB-C power delivery to charge the battery or if it wants to use the data lines.

Unfortunately this negotiation seems taking forever and sometimes even fails.

Solution 1: Strip Power Delivery from Cable

A very simple hack to solve the problem is to strip power delivery. This can be done by using a cable that is not capable of this (like a USB-C to USB-A cable) - or by adding two adapters that strip power delivery from the connection.

I have bought a USB-C plug to USB-A 3.0 socket adapter in which I plugged a USB-A 3.0 to USB-C socket adapter.

I can plug this contraption into my MacBook and the tether cable into it and the issue is completely gone. The camera will be detected immediately afterwards. Occasionally, it will show the battery symbol on waking up, but a half press on the shutter immediately skips this and establishes connection.

MacBook -> Adapter USBC to USBA -> Adapter USBA to USBC -> USBC Cable -> Camera

Adapter construct made from two adapters that can be added to the tethering cable

Solution 2: Use a Battery Adapter

Michael C brought up an interesting theory: If the camera tries to charge the battery instead of transfering data, what if there is a dummy battery adapter like the Canon DR-E6P used, which let's you attach an USB-C charger or power bank directly to the battery compartment?

I can confirm, that this also works.

And that also confirms that my gut feeling that the further you are in your tethering session (the further your battery is depleted), the worse the problem would get.

So, a dummy battery adapter also seems to cure the problem.

Additional Info

I have reported this issue to Canon via Canon Germany. They reported back that they accepted the issue and are sending this to the iternal dev team to look into. I hope that they will make the charging capability switchable via firmware in the future or find another solution.

P.S. On a PC you might be able to disable power delivery to the port you are using in your BIOS. But as I am a Mac user, this is no option for me.