ehhh it really goes much deeper then that. Not all strong EL Nino's are warm winter. We've had some colder than average winter's. Here is a composite of temperature anomalies for weak, moderate, and strong EL Nino winters. ENSO clearly has a strong influence, but there is so much more to it then just ENSO phase and strength. You can see the variation which exists for each season of the same strength magnitude.
Lots get discussed too about structure of the ENSO event and where tropical forcing is positioned, but even with those mixed in the signals aren't entirely clear cut. There are so many other factors and influences.
As I've done composites on an event-by-event basis instead of just grouping together a bunch of years with similar strength or structure it becomes more evident there are many wild cards and ultimately there is always a different feature which is going to be the driver or a key contributor.
Weak:
Moderate:
Strong
Super-strong: