Caveat: I am not a doctor.
First off, don’t undersell the value of blood clotting. You note that blood clotting stops bleeding when you get a cut, but that’s just something that you see externally. Internally, you can suffer much the same bleeding in a variety of places, most notably the GI tract. Clotting helps there too, except that the clotting occurs, the point of bleeding heals, and you’re none the wiser. For small points of trauma, there’s no pain, so it all comes and goes, invisible to you. You might think that having a tiny amount of internal bleeding is no big deal, but remember: it’s slow, but relentless. Having said that, it’s true that I wouldn’t place too much stock in the value of increasing the ability of blood to clot, unless you have a known need to do so. More is not better.
Additionally, many people take a baby aspirin every day to decrease the ability of their blood to clot to avoid strokes and other clot-related problems (deep vein thrombosis, pulmonary embolisms, etc). Aspirin modifies blood platelets such that their ability to clot is decreased (already formed clots are not dissolved, however; that takes a thrombolytic drug, below). When you stop taking aspirin, the resumption of your normal blood clotting ability takes about 10 days, since platelets only last about that long; the effect of the aspirin on the platelets is permanent, so you have to wait for new ones to be made. For this reason, doctors generally ask patients to stop taking their daily aspirin about 10 days before surgery to decrease the risk of hemorrhage.
On the subject of strokes there’s a bit of bad news. Although about 85% of strokes are caused by a thrombus (blood clot) breaking loose and becoming an embolus that becomes lodged in the brain, not all strokes are like that. Hemorrhagic strokes (about 15% of strokes) occur because an artery bursts in the brain. Everything downstream of the break is oxygen starved and dies. Emboli cause brain tissue death through blockage, but hemorrhages cause brain tissue death through leakage. It’s also worth noting that hemorrhagic strokes usually have much more lasting and extensive damage; deaths are also more common. Worse still: it’s not always clear if a stroke is caused by hemorrhage or by an embolism. This is important because embolisms may be treated by clot-buster drugs (thrombolytics), but such drugs will make a hemorrhagic stroke worse since the leakage is prevented from sealing.
Blood clots can dissolve over time; that’s why thrombolytics aren’t always used, particularly if the clot is small. The problem, though, is that damage can occur downstream of the clot in the meanwhile, and there’s always the chance the clot will break loose and wreak more havoc somewhere else (particularly troublesome is a saddle embolism, whereby the embolism rests at the “fork” of a blood vessel, occluding both sides of the fork). As for clogged arteries, there’s more to it than blood clots. Actual damage occurs to the vessel’s wall from plaques formed by cholesterol deposits. This damage is permanent, but progression of the occlusion can be slowed/reversed with cholesterol-lowering drugs (but still, wall damage is permanent). Severe arterial stenosis (narrowing), especially in important areas like coronary arteries, can be treated with stent placement.
BTW, one thing easily checked during a physical is checking the carotid arteries for bruits. A bruit is an abnormal sound heard by placing the stethoscope over the carotid arteries and listening for the characteristic turbulence caused by clogged carotid arteries (bruits can sometimes be heard over other arteries too, e.g., the arteries leading to the kidneys). Next physical, ask the doctor to check.
Take away: more blood clotting is not better, and in many cases may be lessened with drugs to avoid problems related to clots. Cholesterol deposits may be decreased over time with drugs, but underlying blood vessel damage is permanent. Some strokes are caused by clots, and so daily anti-coagulants such as aspirin may help avoid them. Other strokes are caused by vessel leakage and are thus worsened by such drugs. Ask your doctor to listen for clogs.