Discrepancy is that we're mixing (lunar) months and weeks with solar timekeeping, in a solar calendar. These are fundamentally incompatible, so we've gone with cramming the approximate periodicity of the lunar calendar into the solar calendar, while ignoring the fact that we're no longer tracking the moon, and that the weeks don't line up with the year, and the fact that the months are randomly different lengths because they also don't line up and we don't want a weird half-month at the end.

Another potential fix would be having two calendars. A lunar calendar for weeks/months, and a solar calendar for seasons/years.

That’s how units work, fitting the messy natural world into comprehensible numbers. A year is 365 days, except every four, except every 100, except every 400. A month is 30-ish days, and there are 12 of them in a year, because that roughly syncs up the orbits of the Moon and Earth. Except there used to be ten of them (“DECember”), with garbage time filling in the remainder of Earth’s transit around the Sun. A second is something related to Cesium-133, because it’s close to 1/(24x60x60) of a day, because Sumerians chose base 60.

Either all of my old mechanical clocks with moon dial are wrong, or it's 29.5 days.