One reason might be that many of the circuit analyzers also include surge protection. So once the testing is complete, the device can be left connected to protect against surges. They may not be the best form of search protection, because their Joule rating is low, But it’s better than no protection at all.
Please note that I didn't say that's what caused the arcing/overheating. I was commenting on the question of why would anyone leave such a device connected at all times.
