Air traffic would have to work it out as well and why should they be better at it than you?
With the correct (standard?) equipment fitted in the tower, the cross wind is displayed on the same screen as the current "instant" wind, the 2 minute wind, the 10 minute wind, and maximum gust within a similar time period but I forget whether its 2 or 10 minutes.
I think SDB73 is close to the reason of not providing the crosswind on final, as it would become very verbose. Also worth considering is a change in wind direction of a few degrees or a change in strength by a kt or two the crosswind is likely to change more significantly than the mean surface wind. I'll gladly be corrected if I'm talking rubbish here.
However there's nothing wrong with actually asking for the crosswind when you're on final, if it available I'm sure whoever is in the tower will happily provide it.
GW