Depends on one's construct - indicies are composed of elements, some quite different, perhaps orthogonal. These might be factors composing a whole. This begs the question of simple quantification, say by summing factor scores. We return to conceptual issues and recognition of patterns by a summary score; the point being that this isn't wise in all or many instances.

Indices are defined by the theoretical concept that you are trying to measure. In my experience indices tend to be positively correlated as they are purportedly measuring the same concept. In the negatively correlated example you give re Mac and PC use you give the two measures which actually measure the same underlying concept of computer use so both are valid measures of computer use. For example if you take the measure as the amount of time spent on a computer then both combined is not an issue. If however you are looking at type of usage and functionality then perhaps not depending on what you are interested in. It basically boils down to what concept you are trying to measure with the index.

As I understand your question you are basically illustrating the difference between reflective and formative indices - following the terminology used e.g. in the litterature on Structural Equation Modelling. For reflective indices you are assuming that an underlying latent phenomenon is having a (causal) effect on the way your respondents are answering this particular question and for that reason it is reasonable to assume a certain level of correlation between the items. If there is no correlation that means they do not share variance and hence cannot be effects of the same underlying phenomenon. In contrast, formative indices are measures that are based on some sort of derivation of the items in question e.g. the example you give with different types of vegetables. If you are interested in creating an indice measuring the total consumption of vegetables then you cannot assume that there is a correlation between the individual indicators and for that reason no or even negative correlation would not make your indice invalid. Robert DeVellis has a very good discussion of it in his book Scale Development but there are loads of articles on this topic as well.

Interesting question, which you show as a " paradox " in explaining the need to include negative or poorly correlated items knowing that it would report a decrease in reliability. However it is mathematically possible that two items are little or negatively correlated with each other and also have both a positive correlation with a construct that assumes measured by all the items. It is a paradox known in psychometrics : if the items correlate too each other , the reliability is very high, but lower validity . This is because the constructs almost always involve some level of multidimensionality. Thus, a high homogeneity of the items lead to neglect the diversity of aspects of the construct in question. For example, a numeracy test comprises 80 items consisting of sums over another test with 20 additions, 20 subtractions, 20 multiplications and 20 divisions . The first result have higher reliability , but its validity for measuring numeracy will be much smaller than the second .

In short I think you do well to include these items . Only to assess your index, you must also consider the validity , and reliability that you take into consideration