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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My favourite ever gift from friends was that people bought me (last century, long before smartphones) a 24 hour radio synchronized digital alarm clock which knew the day and full date including year. Because while often I know 0600 from 1800, not always, and on a particularly bad day maybe I'm not even sure which week this is. I didn't need the year really, but it felt appropriately completionist, just in case.
I know a retiree that got this as a (sort-of, sort-of-not joke) gift because she mentioned that she sometimes forgets the day:
* https://www.lacrossetechnology.com/products/513-1419v4
(Works best near a window where the radio signal from Colorado can be heard.)