Consider, if there is one person and their chance of failure is P then that is also the chance of failure. If there are two people both with a chance of failure of P then the total chance is the chance that the first person will fail added to the chance that they won't fail but the second person will. I.e. adding the second person makes failure more likely not less.
That is one of the biggest fallacies through the misuse of statistics.
You are talking about the probability of failure by Person 1 (say P1)
or failure by Person 2 (say P2) resulting in a failure, which has no relevance to this incident at Chichester. What you need to be looking at is P1
and P2 resulting in a failure in service. (The probability of TPWS failure is irrelevant in this case, as it does not depend on either Person 1 or Person 2s behaviour, and works independently whatever that probability is, assuming that it has not been tampered with.
So in the simplest form, you need to be looking at P1 (which would be less than 1, ie. always failing) multiplied by P2 (which is also less than 1), therefore the simplest form of mathematical theory would no doubt show that the probability of failure with two people present would be unquestionably lower than with only one person present.
What complicates matters here is that this simplest form assumes that the behaviour of Person 1 and Person 2 are independent, ie. the probability of failure by Person 1 is not affected by the behaviour of Person 2, and vice versa. Of course no matter how much we train ourselves and maintain our discipline, this is rarely the case with human behaviour, so in this case what we need to do is look at the probability of failure by Person 2 conditional on the event of failure by Person 1 (call it Q), and multiply P1 by this conditional probability Q. We only need to consider this case as with train dispatch, the actions are sequential, ie. Person 1 (guard) giving two on the buzzer before Person 2 (driver) performing final checks and applying power.
There are a lot of theories out there which argue that Q is much closer to 1 than P2 is, which is quite understandable. What is not clear is the relationship between P1 x Q (ie. two-person dispatch) and P2 (ie. driver only dispatch), which is part of the reason why this topic always attracts lots of discussion and no conclusive statistical proof either way.
In any case this is still a simplified model, but forms pretty much the fundamental structure of modelling such risks. The difficulty for any practitioner is to come up with accurate figures for these measures.
This does not mean some of your other arguments are wrong, just that I cannot let this go unchallenged as it is one of the biggest mistakes people make in the application of probability, and arguments based on this logic would be null and void. In fact, I quite agree that this isolated incident does not show the inherent danger of anything by itself, although I think most people can agree that someone with only 5 days' training is likely to be less experienced than someone with 6 months' training. I am unable to say whether that makes them more prone to mistakes, although the timing of this incident is really rather unfortunate.