Babies don't receive a lot of vitamin K across the placenta and it is not well expressed in breastmilk - regardless of the mother's diet or vitamin K levels. Formula fed babies receive some oral vitamin K fortification but this is still much less protective than the IM injection.
Vitamin K is a resource that the body (the liver) can store and uses to make clotting factors when necessary. So a single high dose around birth gives the baby enough stores to continue producing clotting factors as needed. Any "excess" of vitamin K will remain in storage - it doesn't cause excessive clotting as some people might imagine it does.
Around 1:10,000 "normal" babies not given supplemental vitamin K will die or be irreversibly disabled by vitamin K deficiency bleeding, which can start spontaneously and unpredictably at days, weeks or even months of age - and many more will have less serious bleeding (that is still very worrying for parents and may need blood transfusion and high doses of vitamin k).
Whilst having a certain type of delivery can increase the chances of early bleeding, the mode of delivery is irrelevant to late onset deficiency bleeding. We are not able to predict which babies will be affected (short of regularly checking all babies clotting function which would be more invasive than vitamin k injection).
It is really difficult in the circumstances of a baby's serious injury or death to come to terms with the baby and family's suffering being completely preventable in most cases. There are almost no cases of haemorrhagic disease of the newborn in babies protected with IM vitamin K. Oral vitamin K is better than nothing, but there is incomplete data on how well absorbed it is (many babies drool or vomit the stuff it seems!) and many busy parents will struggle to complete the full oral schedule. Babies treated with the oral vitamin K are still around 10x more likely to have deficiency bleeding than those treated with the IM injection.
Downsides - while some babies don't seem to mind the injection, most will cry for a short while in my experience. Most babies also cry just as vigorously after they are born, after they vomit, when their nappies are changed... Of course we cannot know what babies subjective experiences are but I do not personally believe the injection is more traumatic than many of the "natural" uncomfortable indignities that newborn babies go through.
Many parents were understandably concerned about a possible link with leukaemia and the IM injection. There has since been further research into this area with no clear evidence of a relationship. If I had to choose between leukaemia and haemorrhagic Vitamin K deficiency bleeding for my child (what an awful thought!) I would choose leukaemia. Children generally survive leukaemia. The risk of leukaemia for me would have to be at least equal to that of death by haemorrhage for Vitamin K not to be worth it. As it, I think that any hypothetical risk of leukaemia is certainly much lower than that of deficiency bleeding.
I am a (pregnant!) junior paediatrician and I spend a lot of time chatting to parents about Vitamin K when they refuse it on the postnatal ward. I hope that I have never bullied a parent and I don't talk about the horror stories (new parents are frightened enough!) but I have seen 1 case of a previously healthy baby die of preventable Vitamin K deficiency bleeding and it is not a devastation I can forget. I would certainly give the IM vitamin K injection to my baby. I don't think most midwives encourage parents enough to have the IM injection or emphasise it is more effective than the oral drops.
It can be difficult to make the decision straight after giving birth! If you have given nothing or given some oral drops but "change your mind" and want the IM injection it's not too late whether the baby is hours or days or weeks old. Insist! Though it will probably be much more difficult to arrange the injection the more time passes.