Well, I would also have a problem with leaving strangers alone in my house. I’m not so worried about the theft of large, valuable things (like TVs or the like), but I am genuinely concerned that our things might be searched through and/or small items could be taken that wouldn’t be noticed immediately – valuable or not. For me, it’s really primarily about having a problem with strangers invading my private living space and being able to search through everything unobserved. "My home" is indeed "my Castle." I don’t want strangers to be in my bedroom or bathroom or other private rooms when I’m not there. That really bothers me. Always has. Even as a teenager, I hated it when someone was in my room unobserved (maybe also because I had bad experiences back then and caught a "friend" reading my diary). Even if I have nothing to hide, it’s a massive intrusion on my privacy for me. And people are naturally curious. Unfortunately, I can’t be sure that my private space will remain private if craftsmen or the like are alone in my house.
If it absolutely couldn’t be helped, I would lock all rooms and only leave open those where the craftsman actually needs to be. Especially bathrooms, bedrooms, and the office are too "intimate" for me to leave strangers there unobserved. If these rooms were locked, I would probably let the craftsmen into my house so, for example, the cleaner could get water from the utility room or kitchen if there is no way for my husband or me to be there at the same time. Strangely, I’m not so sensitive about the living room/kitchen/utility room. The other rooms would then be locked, and I would also lock up money and so on in these rooms (of course, no guarantee either, but I’d not really expect the craftsmen to have such criminal energy as to break down doors). But if work had to be done in the "private rooms," I would insist that my husband or I be present at the same time.
We don’t have practical "experience" with this yet. Our house is still under construction. This hasn’t happened in our apartment yet. We’ll see how we actually handle it later.