Why can't we use a screen to prevent the clot from going to my lungs?

You can. But the more effective and safer treatment is a blood thinner (warfarin or enoxaparin for example). "screens" or ivc filters are reserved for when blood thinners cannot be used, or when a clot develops in spite of a blood thinner. They do not treat the underlying problem (clots in the veins). They only prevent the clot from migrating.
You can. An ivc filter is a screen that goes in the vena cava and can block clots from moving from the leg viens to the lungs. There are side affect and short and long term risks with putting in ivc filters, thus it is a case by case decision whether to put them in.