Mathematically, why does rounding 5 up bias data (as opposed to rounding to nearest even)?

Randomness reduces bias. When x is rounded up, you'll always get a number that's at least as large as x. Whenever x is rounded to the nearest even, it could be rounded up or down. Consider the simplest of statistics in the simplest scenario: the mean and sum, where the data is positive numbers. Scenario 1: Every time you round up, the mean and sum will systematically increase. Scenario 2: Every time you round to the nearest even, the mean and sum could increase, decrease, or stay the same. in Scenario 1, the outcomes are always biased upwards. In scenario 2, the bias is more "random". Of course, there are statistics like estimators for non-linear regressions and other wonky things I can't imagine which defy my simple example. But in general, if there is going to be some error introduced into the data that you don't know about (and therefore can't fix), it's better if the outcome of interest isn't systematically affected. (However, if you know about the error, then it's better to have it be systematic so you can systematically correct it!)