I agree with Suz that six months is a long time for this to have gone on. Are there other changes in her life (starting preschool? New sibling? Marriage issues? Anything?) that might be the deeper root of all this fear and she's channeling it into the doll issue?
Or if you have given her lots of attention over the doll thing and talked and talked it over with her, she might have learned that it gets her your attention (especially nice for her if she has siblings). So don't keep hashing it over verbally.
However, if you have been saying over and over, "No, we won't be with you all the time" and it's become a thing where she sees it as her mission now to get you to be with her -- then do it, and let this burn out.
Now that she can''t "un-see" the video: Take her seriously and just go with this for right now. She will not be this fearful forever, and right now I'd give her what she needs, and what she needs is reassurance that she can only get from having an adult nearby. Please don't try telling her,,"You're too big to have us around all the time" or "You didn't need to sleep with us before so you don't now" or anything else that dismisses her fear or gives her the message that she's somehow being babyish and is wrong to want comfort. She is still too young really to control her imagination or to take it in if you tell her, "Dolls don't really move or talk" -- you can say it all day long, AND she can nod and say, "Yes, I understand" all day long, but that does not mean her mind is able to overcome this fear right now.
If she''s sleeping in your room: Get her to sleep in her room but you or dad stays until she goes to sleep, if that's what it takes for now. I know, adults long for the time that kids sleep in their own rooms without parents there at all, but I'd do it temporarily if that gets her past this. You will not be doing it forever, no matter whether it feels that way at the moment.
If she starts to talk about these dolls, reassure her but briefly and then distract her with another subject or something specific to do with her hands--"Let's get out the crayons and that coloring book you love" or whatever. Gradually she'll be more OK as you move a little farther away while she colors or plays with toys. You might also, if she is really wound up about dolls, ask if she wants to have a special time when she and you gather up all her dolls and "retire" them temporarily to somewhere -- don't throw them out as she's likely to want them again later, but give them a vacation. If she is OK with stuffed animals but not dolls right now, maybe one special stuffed animal can be designated by her as the one to go places with her and keep an eye on her. Also check her whole room with her -- are there things in the room that seem OK by daylight but weird by the light of her nightlight and cast shadows that now seem scary? Move those things away to show her they're only shadows. If she follows you to the bathroom, let her -- but do not let her in; have her stand outside the door and talk with her so she hears your voice (and so you maintain some boundaries but she still knows you're there). Don't ever over-talk the problem or spend a lot of time discussing the fear or the moving dolls idea, but do acknowledge and then distract immediately.
I know it's driving the adults "insane" as you put it but this type of clingy fear is pretty typical when a kid has had a scare like she had. Please don't push her away. That may seem like the adult thing to do, or the way to make her more independent and less fearful, but if you dismiss her fear or push her away (or worst of all--make fun of her fears like some adults do, believe it or not)--she will NOT get the message that she needs to grow up. She will only get the message that she can't rely on mom and dad to listen to her when she's scared and to provide comfort without criticism.
I'm not saying you or your husband are doing any of those things since you don't say how you're handling this right now! I'm just noting that it's easy for us adults to tell young kids to "stop being babyish" or "You know dolls can't move!" and those things send a young kid a message very different from what we might intend.
I think you already know that she should not be on YouTube at all, ever, even with you there -- It's far too easy to be watching some innocent video and then the next video in that "list" suddenly starts playing and it's not appropriate at all. I can totally see how a search on "dolls" went instantly from a sweet doll video to some scary movie clip etc. In fact at her age she should only do very limited screen time of any form for many reasons -- so her own imagination can develop, so she's not overstimulated by the flickering images (that really does happen -- check the American Academy of Pediatrics' web site about kids and screen time) and so she doesn't end up seeing stuff she shouldn't.