"Run every day" is a default setting we can choose for programming. For example, my computer is set to block pop-ups as its default setting. However, if needed, I can choose to override the default setting based on cirumstantial need. Similarly, with running, a default setting of "run every day" is more conducive to long term consistency than building in a day to "lick our wounds" from the previous six or seven days.
That's far different than running streaks ("never take a day off"), which are more of a circus act than a repeatable training approach.
If someone needs to take a day off every single week to avoid injury, they're not in control of their training...they're surviving it. Unfortunately, in endurance sports, we often glorify survivalism and the coaches who perpetuate it.
There's no doubt that you can thrive at the highest levels of sport with a day off per week...but that doesn't make it the optimal formual for each individual.
If we want to talk anecdotes, Michael Phelps in his prime went a five year stretch where he missed something like two days . Now, we might say that Michael Phelps simply survived that training better than anyone else could (that's a perfectly legitimate question). But a closer look reveals that he actually did less overall yardage per week than many comparable swimmers, which facilitated the consistency needed to swim seven days per week and swim about 250 more days in a five year stretch than others who took a day off per week. The North Baltimore "hell weeks" over Christmas break were actually more "tame" than other elite programs' hell weeks.
In sum, a "run every day" default setting (which is different than the code for "never take a day off") is more about an overall approach to facilitate long term consistency rather than the odd obsessiveness of a running streak. The real justification for running every day has more to do with neurology than physiology, when we look at elite performers in fields outside of sports (musicians, chess masters, etc).
Doctor of Physical Therapy
Tinman athlete since 2003