That someone would let an agent run on it at any time, read what's there, and possibly modify the code or interrupt a process, without even informing me, is what I find problematic, to say the least.
The employer can't enter your machine to look at your personal info just because they feel like it, but they can access it for work related reasons to access work related info, if they have just cause.
Claiming the employer in the EU has no right to access your machine is plain false. It's the company's machine, on the company's network, inside the company's premises, it can definitely access it for a business use case. Now if you happen to put your private info there because you're sloppy, then it's on you. How hard is it to avoid putting personal info on company property?
I expect that I can be relocated to another office with a month of advance notice. Also cleanup, fire security check and fixing water leakage while I'm not there.
I don't expect that someone else will provide the keys of "my" office to the biology department when they have to keep a few sheeps during the weekend.
It's not your production machine. It's the company's production machine. Always expect the company to do whatever it wants with its machine, and use the machine accordingly.