Personally I find it more handy to know day (of week) than date (of month), but I don't think I've every seen a watch that does day-only, usually either date-only or day-date.
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But most people always know exactly what day of the week it is, but rarely know the date. (The only time I'm ever unaware of the day is on vacation occasionally.)
And the date is needed basically every time I sign anything on paper, as you almost always put a date next to any signature -- whether you're writing a check, filling out forms at the doctor's office, signing consent forms for an activity, and so forth.