Bivariate correlations are not the only way to diagnose singularity in your dataset. You may want to see what happens when you predict each of your variables from the set of other variables. A common mistake in psychology datasets is to include scale scores as well as items where the scale scores are a function of the items.
KMO relates to properties of the overall correlation matrix. You could for example add a random variable unrelated to any of the other variables and still get a decent overall KMO.
In general, there are many reasons to justify removal of a variable from a factor analysis. This is a bit of an art.
If you are truly doing factor analysis and you have a variable that is unrelated to the other variables then this may be a reason for item removal. Alternatively, it may flag that you should have had more variables measuring the construct that the variable relates to.
If you are doing PCA, then you may just be concerned with data reduction, and as such an independent item may not be a problem.
You should make an assessment of the degree of independence of the variable from the others. Think conceptually about whether you think this independent variance is substantively interesting or whether it is because the variable failed to measure anything interesting.
In general, you want to integrate content knowledge and the statistics to make a reasoned judgement.