Having survived a pulmonary embolism from ACL reconstructive surgery, it is not the "code red" you make it out to be. Assuming the little clam shell on the sidelines doesn't have a full setup to perform an angiogram, and considering his pants were still on (because they would have had to remove those to shave his nuts and poke the catheter up through the vein in his groin to go through the heart to get to his lungs), they didn't "find" a blood clot right there on the sidelines. Besides, if a clot was suspected, they would have stuck him immediately with a heparin IV drip. Been there. Done that.
Timing is of the essence, but it is not an emergency C-section git 'er done in 30 minutes or less kind of ordeal. The medical staff that treated me moved efficiently, but at no time did they act like the building was on fire and I was wearing gasoline soaked clothing.
Since I am a PE survivor, I am much more susceptible to clots in the future, and I didn't hesitate to get vaccinated. The probability of that outcome happening as a result was so small as to be negligible.