Coumadin/warfarin specifically inhibit a natural pathway in the body for clotting. From http://www.rxlist.com/cgi/generic/warfarin.htm:
"COUMADIN (crystalline warfarin sodium) is an anticoagulant which acts by inhibiting vitamin K-dependent coagulation factors." (don't miss the black box warning--it's a nice one!!!).
This is different than many natural compounds that basically make the blood "less sticky" (as in reduced inflammation). "Less sticky" blood can still coagulate just fine, whereas inhibited clotting mechanisms screw up everything everywhere.
Proper coagulation is essential in health. Note that all of the compounds you mentioned are generally associated with improving health when taken in proper doses. Any honest healthcare practitioner will tell you that coumadin/warfarin kills people. Many still stroke out, even after suffering for many years on that drug. There are way more mechanisms to coagulation than just the Vitamin K pathway. Also, people on that drug are basically told to eat NO green vegetables!!! How healthy is that?
I'm done ranting. I hate that drug.