Of the two categories with injury ties, Medical is the one you could skimp on if you wanted. Here's the rationale:
Training PREVENTS injuries in the first place. The higher your training budget, the less likely you are to see injuries of any kind to your players.
Medical limits severity of injuries. So, once you do get an injury of some kind, the duration and ratings hit and ratings recovery rate are all effected by the Medical budget.
As they say, an ounce of prevention is worth a pound of cure.
So, the owners who swear by a $0M medical budget will espouse the importance of only recruiting players with high health ratings. Since the injuries you do get are likely to be severe, you really want them to be as few and far between as possible. High health would be especially important for any prospects (players you need to stay healthy for several years in order to give you value) or when signing an FA to a multi-year deal.