I do not consider that a good idea.
First of all, you should listen to instructions issued to the aircraft ahead of you to have a general picture of the situation, ATIS may also be a source, although sometimes rather old I admit. Checking the wind when receiving the landing clearance is rather late...
Providing the crosswind component does not make sense if you have different runways (two components for each runway, ATIS gets longer and longer, more confusion...). Even more, you do not have to make an exact caculation, if you are not close to your limit, since wind direction changes every second anyway.
Bernhard