Fibrinolysis is the destruction of fibrin. Fibrin is the insoluble protein in clotted blood. In order for blood to clot, sufficient amounts of fibrin (and other things like calcium) must be present.
I respectfully disagree with the second part of your question because intravascular blood very often does in fact remain clotted inside the lumen of blood vessels. In fact, this is the etiology of many disease processes. DVT's (deep venous thrombosis) is a good example of this phenomenon. Clots (thrombi) develop in the deep veins of the legs for varying reasons. This produces pain, swelling, heat, and decreased range of motion to the affected extremity.
Another disease process that is due to clotted blood in the blood vessels is called acute coronary syndrome. Thrombi in the coronary arteries cause a multitude of signs and symptoms and are a major cause of sudden cardiac death.