They should try to take some of the individual differences in the autonomous nervous system functioning into account by asking questions about the relevant issues and determining a personal baseline to which the results are compared. If you have e.g. an underlying undiagnosed disease that raises your blood pressure it would be present both during the control questions and the case relevant questions.
But still, anyone can get more nervous when asked "Did you murder your boss?" than "Is your name Donjeta?". It's tricky when it's obvious which are the important questions. I think they should concentrate on developing guilty knowledge tests. It's a variation of LDT in which you are tested on your reactions on stuff that presumably only the perp knows. Innocent people who don't know which are the important items and which are the distractor items would be expected to be equally nervous throughout. But it would be rather hard to do in Kyron's case in this case because the police would have to be sure what happened first.
I agree that guilty knowledge polygraphs are more promising than the traditional type but they are still far from conclusive (or, for me personally, convincing).
A guilty knowledge polygraph is a variation on an old interrogation technique whereby the subject makes statements that, in theory, only LE and the perpetrator would know.
The problem with this in previous cases is controlling what every single LE and witness says. Some subjects are extremely good at figuring out what LE wants to hear and some LE are amazingly garrulous.
An example of a suspect who was extremely good at figuring out what LE wanted to hear was Henry Lee Lucas. He probably only killed three people (one of which could be argued to have been self defence) and he was definitely not the sharpest knife in the block. He started confessing to murders and pretty soon, LE over most of the US was interested in him. He ended up confessing to over 600 murders.
Oddly enough, his confessions were always consistent with the details known by the LE person interviewing him about them.
It was later shown that in the vast majority of his confessions, the "secret" details had been inadvertently fed to him by the investigators questioning him.
So, as a hypothetical example, say a suspect is taken to the police station to be questioned. On his way to the station, the police officer driving the car tells him that the victim was murdered via three knife slashes across the throat. The subject asks lots of questions and the slip of the tongue is forgotten by the officer involved.
Then at the police station, while being questioned, the suspect refers to the victim as having their throat cut. The interrogators, none of whom were in the car for the ride to the station, believe this is an example of guilty knowledge.
If they use a polygraph and bring out a series of potential weapons, a factually innocent person under those circumstances may well have an emotional response to being shown a knife, which they would infer had been used in the crime.
Finally, as you say, there are cases like Kyron's where the investigators probably don't have any physical items known to be used in the crime.