I think we agree in practice but not in theory. Without any doubt, the term “(statistically) significant” is misunderstood (and as such meaningless at best and harmful at worst) in practice. People (wrongly) think that significance is a proof of the existence of an effect, if not a relevant effect. That led to many bad conclusions and decisions, even to absurd procedures (like formal tests to justify assumptions, or step-wise selection procedures). Seeing what is done, practically, lets me absolutely agree with your point-of-view, that “significance” is meaningless and harmful.
However 
, the theoretical concept still sounds sensible to me. The p-value measures the “unexpectedness of more extreme (sufficient) test statistic under H0 (and the statistical model) than the one calculated from the observed data”. This lengthy description is condensed in the word “significance”, and in the common way of using a language it is ok and understandable, I think, to say that the data “is significant” if its significance is so high that it seems obvious that data + model + H0 don’t go together. Iff (i) the data were collected with greatest care and (ii) the model is described correctly, then and only then a high significance indicates that the information in the data considerably discredits H0 (and the information is sufficient to interpret “at which side of H0” the data is). The practical point is, for sure, that it’s impossible to have a “correct model” (it’s a model!). So, strictly following the first part of George Box’s citation about models (“All models are wrong”), the whole concept is wrong, and in this line one may conclude that it’s meaningless and even harmful. But the second part (“but some are useful”) supports my view: if the model is not too bad, and the data is not too much, the significance (p-value) is a useful measure to judge the information of the data regarding H0.
I am curious how you think about this (in other words: I am happy to learn where I am wrong).