that very small p-value indicates it does not depart significantly from a uniform distribution

I think you got the interpretation wrong. Normally, one would reject the null hypothesis (=uniform distribution here) if the p-value is less than 0.05 or 0.01, corresponding to 5% or 1% significance level (i.e. the error probability of incorrectly rejecting the null hypothesis despite it being true). In other words, the deviations from a uniform distribution are statistically highly significant.

In reply to Re^2: [OT]: Statistical significance? by Anonyrnous Monk
in thread [OT]: Statistical significance? by BrowserUk

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