Why does the day of the week follow a mod-7 pattern?

Because the week is a fixed 7-day cycle — moving forward N days shifts the weekday by N mod 7.

Weekdays repeat every 7, so only the remainder of any day-count matters: 7 days later is the same weekday, 8 days later is +1, 30 days later is +2 (since 30 = 4·7 + 2). Doomsday exploits this by never tracking the actual number of days — it works entirely with those small remainders, which is what keeps the mental arithmetic tiny.