I hope there are plenty of comments answering this question, as I only know my opinion and don't want it to stand alone.
I don't think it is a matter of readiness to bite or taking awhile to warm up. I think it has to do with the LGDs assessment of the situation. I've heard of dogs who always place themselves between intruder and flock member (human or 4 legged). When I drove to Idaho to get my pup, I had never so much as laid eyes upon a komondor. I had only what I could learn from
online research. I read a story of a komondor holding a burglar on the ground with jaws on the intruder's neck. The dog was absolutely certain this individual was up to no good, and was reluctant to let the intruder up with the owner's urging, and the police present. I think you could have an aggressive LGD, but I think it would be rare. Inappropriate aggression would be cause for the other LGDs to intervene, likely kill the inappropriately aggressive dog. For a thousand generations, the ones with an aggressive nature were selected against by their own packmates. During those years, the adult herbivores butted and chastised the LGDs as pups, helped with their training. Aggression was bred out of them. Affection (for the flock) was selected, protectiveness was selected, powerful build, savvy fighter, cooperative with the other LGDs optimized all these many long years.
Most people I know vow their LGDs are affectionate gentle and loving. I always HOPE my LGD would intervene if an intruder came, and I was not safe, but I have not found out, and really guess I'd rather never find out. She did allow herself to be dognapped last summer, went with what appeared to her to be a nice boy, I guess, got in his car and rode away.... I did get her back, but that's another story.
Read everything you can get your hands on about LGDs, and imagine them in the conditions in which their breeds differentiated from the other canine races. They lived in the flock of herbivores, they did independent patrols, worked independently or in teams, without human direction. All canines are pack animals, LGDs pack was associated with a specific flock of herbivores, they bonded to those particular herbivores, they imprinted on a set of conditions of how things ought to be, and the rest of their lives, they maintained that set of conditions or died trying.
The LGD has a sense of responsibility and will be true to how ever the individual LGD perceives her responsibility. They monitor the situation, place themselves strategically, at the ready, will respond with what ever force they deem necessary. I think if they perceive something "off" about a particular individual, they will never warm up to him, might tolerate his presence, but be watchful when ever the individual is present. As long as that 'off" individual human didn't try any funny business, it is unlikely the LGD would attack or bite.
I don't see them as biting. Little yappy dogs bite (I have one of those too). Little dogs are worriers, and it makes sense. They are fragile creatures, an accidental contact with a moving foot could injure them, they yip and snap to keep their way clear. If I step over my little dog, she is alarmed, she jumps up, she runs under where my foot will come down. (staying still would have kept her out of harms way, but she is an alarmist). On the other hand the LGD is large and self assured. It would take a lot to injure her, and she knows it. She is a benevolent giant. You can step over her, lie on her... so what, she doesn't care, she likes the contact. If you fall on her she'll just look up to try to figure out wha'? But she'll forgive you the inconvenience you caused her. They don't bite or nip or snap, they're not trying to bluff you or frighten you. They're quietly capable of stopping you in your tracks, with almost no effort at all, and they know it.
I look forward to reading the opinions of others on this.
Thekla