I think you are misreading the relevant passage here.
Since you do not specify exact excerpt(s), I take that by "implicit assumption" you refer to the equation (2) (application of a ReLU) and the corresponding text explanation (bold emphasis mine):
We apply a ReLU to the linear combination of maps because we are only interested in the features that have a positive influence on the class of interest, i.e. pixels whose intensity should be increased in order to increase $y^c$. Negative pixels are likely to belong to other categories in the image. As expected, without this ReLU, localization maps sometimes highlight more than just the desired class and perform worse at localization.
The first thing to notice here is that this choice is not at all about activations close to zero, as you seem to believe, but about negative ones; and since negative activations are indeed likely to belong to other categories/classes than the one being "explained" at a given trial, it is very natural to exclude them using a ReLU.
Grad-CAM maps are essentially localization ones; this is apparent already from the paper abstract (emphasis mine):
Our approach – Gradient-weighted Class Activation Mapping (Grad-CAM), uses the gradients of any target concept (say ‘dog’ in a classification network or a sequence of words in captioning network) flowing into the final convolutional layer to produce a coarse localization map highlighting the important regions in the image for predicting the concept.
and they are even occasionally referred to as "Grad-CAM localizations" (e.g. in the caption of Fig. 14); taking a standard example figure from the paper, e.g. this part of Fig. 1:
it is hard to see how including the negative values of the map (i.e. removing the thresholding imposed by the ReLU) would not lead to maps that include irrelevant parts of the image, hence resulting in a worse localization.
A general remark, largely irrelevant to the exact issue: while your claim that
After all, neurons have biases, and a bias can arbitrarily shift the reference point, and hence what 0 means
is correct as long as we treat the network as an arbitrary mathematical model, we can no longer treat a trained network as such. For a trained network (which Grad-CAM is all about), the exact values of both biases & weights matter, and we cannot transform them arbitrarily.