If you can't land without ATC supplying a crosswind component for you, are you never going to fly into or out of an uncontrolled airfield? Are you never going to have to make a forced landing?
A lot of ATC units don't have the equipment to display the resolved winds and, given that wind is a continuously variable fluid, would have to spend all their time doing maths instead of controlling aircraft. We could always slow down the movement rate to one take-off or landing every 5 mins or so, I suppose, to give us the time?
Try spending some time taking random winds & resolving them in your head, hen doing the maths & seeing how close you are. You will get better with practice. There are also various accessories (eg flight rulers, fwizz-wheels etc) with basic resolution charts printed on them, and I can recommend the RAF Flight Information Publication (FLIP) to you - loads of conversion charts, climb/descent gradients, wind resolution charts, flight-planning & emergency info. It's a little yellow book that doesn't at all mind being bent out of shape & being stuffed into a flight-bag!