I'm currently re-watching season one Voyager and it is very clear in those early episodes that the Doctor is not able to switch himself on or off at will, even when he has important work he wants to get on with. It is actually a plot point in the first few episodes, which Kes has to argue on his behalf to Captain Janeway, because it hadn't occurred to anyone in the crew. So no, the out-of-the-box Doctor in Voyager would not, in fact, be able to report a murder he had witnessed, if the murderer had switched him off, because he would not be able to activate himself in order to do so. The ability to switch himself on or off had to be given to him, specifically added to his programme, when he asked for it, because it hadn't occurred to anyone that he might need such an ability. And even then Kes had to fight for it.Nothing we see in Voyager suggests an out-of-the-box Picardo would fail to report a murder .
Rios's holograms are pretty much straight out of the box. Whatever fiddling he has done with their programming has not improved it. The show has demonstrated that they are not able to switch themselves on or off at will, but rather activate in response to specific situations and commands. Agnes switched the EMH off and after that nothing tripped his activation protocol until she collapsed, therefore he wasn't able to report what he had seen.
ETA it is worth adding that if Beautiful Flower had been a Data clone, there were more people on the ibn Majid than just Rios - someone on that crew would have recognised him and commented on the resemblance, and Rios would most likely have heard about it, even if he didn't clock the likeness himself.
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